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Sublinear function

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In <a href="Linear_algebra" title="Linear algebra">linear algebra</a>, a sublinear function (or <a href="Functional_(mathematics)" title="Functional (mathematics)">functional</a> as is more often used in <a href="Functional_analysis" title="Functional analysis">functional analysis</a>), also called a quasi-seminorm or a Banach functional, on a <a href="Vector_space" title="Vector space">vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is a <a href="Real_number" title="Real number">real</a>-valued <a href="Function_(mathematics)" title="Function (mathematics)">function</a> with only some of the properties of a <a href="Seminorm" title="Seminorm">seminorm</a>. Unlike seminorms, a sublinear function does not have to be <a href="Nonnegative" class="mw-redirect" title="Nonnegative">nonnegative</a>-valued and also does not have to be <a href="Absolutely_homogeneous" class="mw-redirect" title="Absolutely homogeneous">absolutely homogeneous</a>. Seminorms are themselves abstractions of the more well known notion of <a href="Norm_(mathematics)" title="Norm (mathematics)">norms</a>, where a seminorm has all the defining properties of a norm except that it is not required to map non-zero vectors to non-zero values.

In <a href="Functional_analysis" title="Functional analysis">functional analysis</a> the name Banach functional is sometimes used, reflecting that they are most commonly used when applying a general formulation of the <a href="Hahn%E2%80%93Banach_theorem" title="Hahn–Banach theorem">Hahn–Banach theorem</a>.

The notion of a sublinear function was introduced by <a href="Stefan_Banach" title="Stefan Banach">Stefan Banach</a> when he proved his version of the <a href="Hahn-Banach_theorem" class="mw-redirect" title="Hahn-Banach theorem">Hahn-Banach theorem</a>.<a href="#cite_note-FOOTNOTENariciBeckenstein2011177–220-1">[1]</a>

There is also a different notion in <a href="Computer_science" title="Computer science">computer science</a>, described below, that also goes by the name "sublinear function."

<div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2></div>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> be a <a href="Vector_space" title="Vector space">vector space</a> over a field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {K} ,}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {K} ,}</annotation>

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</math><img src="./07453cff88535372e3ebb320517acf77be68529d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.455ex; height:2.509ex;" alt="{\displaystyle \mathbb {K} ,}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {K} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {K} }</annotation>

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</math><img src="./1848c435e64864e9ad4efa7e46bd6bc900c35c99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {K} }" loading="lazy"> is either the <a href="Real_number" title="Real number">real numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

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</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"> or <a href="Complex_number" title="Complex number">complex numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} .}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} .}</annotation>

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</math><img src="./8f4d5d3ec97eee8b915d3b14d3fb38579ee639d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} .}" loading="lazy">

A real-valued function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:X\to \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle p:X\to \mathbb {R} }</annotation>

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</math><img src="./c8540db038c97641c58fdf2dbe344cc730d8b441.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.468ex; height:2.509ex;" alt="{\displaystyle p:X\to \mathbb {R} }" loading="lazy"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is called a <style data-mw-deduplicate="TemplateStyles:r1238216509">

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</style>sublinear function (or a sublinear <a href="Functional_(mathematics)" title="Functional (mathematics)">functional</a> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {K} =\mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {K} =\mathbb {R} }</annotation>

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</math><img src="./aaa0cac96c9853f0cba61605679cf8963983b5d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.585ex; height:2.176ex;" alt="{\displaystyle \mathbb {K} =\mathbb {R} }" loading="lazy">), and also sometimes called a quasi-seminorm or a Banach functional, if it has these two properties:<a href="#cite_note-FOOTNOTENariciBeckenstein2011177–220-1">[1]</a>

<ol>

<li><a href="Positive_homogeneity" class="mw-redirect" title="Positive homogeneity">Positive homogeneity</a>/<a href="Nonnegative_homogeneity" class="mw-redirect" title="Nonnegative homogeneity">Nonnegative homogeneity</a>:<a href="#cite_note-FOOTNOTESchechter1996313–315-2">[2]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(rx)=rp(x)}">

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<annotation encoding="application/x-tex">{\displaystyle p(rx)=rp(x)}</annotation>

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</math><img src="./ac3c12bca9edb1a463577e9b747d5e343f389223.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:13.902ex; height:2.843ex;" alt="{\displaystyle p(rx)=rp(x)}" loading="lazy"> for all real <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\geq 0}">

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<annotation encoding="application/x-tex">{\displaystyle r\geq 0}</annotation>

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</math><img src="./aa96c19954fcda2695f988938ccf091d2bc2bbae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.31ex; height:2.343ex;" alt="{\displaystyle r\geq 0}" loading="lazy"> and all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X.}">

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<annotation encoding="application/x-tex">{\displaystyle x\in X.}</annotation>

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</math><img src="./0deab6a01578b5b543b772df12dc0d2c593cc924.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.797ex; height:2.176ex;" alt="{\displaystyle x\in X.}" loading="lazy">

<ul><li>This condition holds if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(rx)=rp(x)}">

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<annotation encoding="application/x-tex">{\displaystyle p(rx)=rp(x)}</annotation>

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</math><img src="./ac3c12bca9edb1a463577e9b747d5e343f389223.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:13.902ex; height:2.843ex;" alt="{\displaystyle p(rx)=rp(x)}" loading="lazy"> for all positive real <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r>0}">

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<annotation encoding="application/x-tex">{\displaystyle r>0}</annotation>

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</math><img src="./23cbbcd53bd13620bc53490e3eec42790850b452.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.176ex;" alt="{\displaystyle r>0}" loading="lazy"> and all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X.}">

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<annotation encoding="application/x-tex">{\displaystyle x\in X.}</annotation>

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</math><img src="./0deab6a01578b5b543b772df12dc0d2c593cc924.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.797ex; height:2.176ex;" alt="{\displaystyle x\in X.}" loading="lazy"></li></ul>

</li><li><a href="Subadditivity" title="Subadditivity">Subadditivity</a>/<a href="Triangle_inequality" title="Triangle inequality">Triangle inequality</a>:<a href="#cite_note-FOOTNOTESchechter1996313–315-2">[2]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x+y)\leq p(x)+p(y)}">

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<annotation encoding="application/x-tex">{\displaystyle p(x+y)\leq p(x)+p(y)}</annotation>

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</math><img src="./b4c62346e0a4a5625be6fcc47ae8c7183de5041f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:22.775ex; height:2.843ex;" alt="{\displaystyle p(x+y)\leq p(x)+p(y)}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in X.}">

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<annotation encoding="application/x-tex">{\displaystyle x,y\in X.}</annotation>

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</math><img src="./6a440b9e84ab4269d8f780f8e8a8c9aaf3cd832f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.987ex; height:2.509ex;" alt="{\displaystyle x,y\in X.}" loading="lazy">

<ul><li>This subadditivity condition requires <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> to be real-valued.</li></ul>

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A function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:X\to \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle p:X\to \mathbb {R} }</annotation>

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</math><img src="./c8540db038c97641c58fdf2dbe344cc730d8b441.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.468ex; height:2.509ex;" alt="{\displaystyle p:X\to \mathbb {R} }" loading="lazy"> is called positive<a href="#cite_note-FOOTNOTENariciBeckenstein2011120–121-3">[3]</a> or nonnegative if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)\geq 0}">

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<annotation encoding="application/x-tex">{\displaystyle p(x)\geq 0}</annotation>

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</math><img src="./2d733d94db74e18da00415b0ef96a820d370d874.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:8.659ex; height:2.843ex;" alt="{\displaystyle p(x)\geq 0}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X,}">

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<annotation encoding="application/x-tex">{\displaystyle x\in X,}</annotation>

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</math><img src="./e3ebdb0a09f0721ccdd0b779e0a21caf386be82a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.797ex; height:2.509ex;" alt="{\displaystyle x\in X,}" loading="lazy"> although some authors<a href="#cite_note-FOOTNOTEKubrusly2011200-4">[4]</a> define positive to instead mean that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)\neq 0}">

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<annotation encoding="application/x-tex">{\displaystyle p(x)\neq 0}</annotation>

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</math><img src="./6376614de7c9124850da29ee7f477d0deb5af9c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:8.659ex; height:2.843ex;" alt="{\displaystyle p(x)\neq 0}" loading="lazy"> whenever <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\neq 0;}">

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<annotation encoding="application/x-tex">{\displaystyle x\neq 0;}</annotation>

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</math><img src="./77fbee59e2db32fddccc0327e5582d6a31d90d92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.237ex; height:2.676ex;" alt="{\displaystyle x\neq 0;}" loading="lazy"> these definitions are not equivalent.

It is a symmetric function if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(-x)=p(x)}">

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<annotation encoding="application/x-tex">{\displaystyle p(-x)=p(x)}</annotation>

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</math><img src="./6e25594f5a6ea3ae62f0545ad97ff30419766102.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:13.613ex; height:2.843ex;" alt="{\displaystyle p(-x)=p(x)}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X.}">

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<annotation encoding="application/x-tex">{\displaystyle x\in X.}</annotation>

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</math><img src="./0deab6a01578b5b543b772df12dc0d2c593cc924.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.797ex; height:2.176ex;" alt="{\displaystyle x\in X.}" loading="lazy">

Every subadditive symmetric function is necessarily nonnegative.<a href="#cite_note-SubadditiveSymmetricIsNonnegative-5">[proof 1]</a>

A sublinear function on a real vector space is <a href="#symmetric_function">symmetric</a> if and only if it is a <a href="Seminorm" title="Seminorm">seminorm</a>.

A sublinear function on a real or complex vector space is a seminorm if and only if it is a <a href="Balanced_function" class="mw-redirect" title="Balanced function">balanced function</a> or equivalently, if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(ux)\leq p(x)}">

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<annotation encoding="application/x-tex">{\displaystyle p(ux)\leq p(x)}</annotation>

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</math><img src="./d38153f37ed7c4ee142b2ade6f7556b7bafbf402.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:13.134ex; height:2.843ex;" alt="{\displaystyle p(ux)\leq p(x)}" loading="lazy"> for every <a href="Unit_length" class="mw-redirect" title="Unit length">unit length</a> scalar <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

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<annotation encoding="application/x-tex">{\displaystyle u}</annotation>

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</math><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> (satisfying <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |u|=1}">

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<annotation encoding="application/x-tex">{\displaystyle |u|=1}</annotation>

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</math><img src="./d032403a63bd4b85a69d19fd96192bbb79f2ffeb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.884ex; height:2.843ex;" alt="{\displaystyle |u|=1}" loading="lazy">) and every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X.}">

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<annotation encoding="application/x-tex">{\displaystyle x\in X.}</annotation>

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</math><img src="./0deab6a01578b5b543b772df12dc0d2c593cc924.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.797ex; height:2.176ex;" alt="{\displaystyle x\in X.}" loading="lazy">

The set of all sublinear functions on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}">

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<annotation encoding="application/x-tex">{\displaystyle X,}</annotation>

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</math><img src="./09ba32eeb405f7f5f2bac1eb12987c47d2fd42df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" loading="lazy"> denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{\#},}">

<msup>

</mrow>

</msup>

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<annotation encoding="application/x-tex">{\displaystyle X^{\#},}</annotation>

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</math><img src="./5332a4a460b711212ca89b7045c577e2788e556e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.245ex; height:3.009ex;" alt="{\displaystyle X^{\#},}" loading="lazy"> can be <a href="Partial_order" class="mw-redirect" title="Partial order">partially ordered</a> by declaring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\leq q}">

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<annotation encoding="application/x-tex">{\displaystyle p\leq q}</annotation>

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</math><img src="./9e46e733ff7a6d62f7685d4c20e4a467e984dfdf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.427ex; height:2.343ex;" alt="{\displaystyle p\leq q}" loading="lazy"> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)\leq q(x)}">

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<annotation encoding="application/x-tex">{\displaystyle p(x)\leq q(x)}</annotation>

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</math><img src="./30a407362d50c9dcc7102567c99ec2e5892f1c20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:11.705ex; height:2.843ex;" alt="{\displaystyle p(x)\leq q(x)}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X.}">

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<annotation encoding="application/x-tex">{\displaystyle x\in X.}</annotation>

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</math><img src="./0deab6a01578b5b543b772df12dc0d2c593cc924.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.797ex; height:2.176ex;" alt="{\displaystyle x\in X.}" loading="lazy">

A sublinear function is called minimal if it is a <a href="Minimal_element" class="mw-redirect" title="Minimal element">minimal element</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{\#}}">

<msup>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle X^{\#}}</annotation>

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</math><img src="./7197388bb622582f173a2c78284e849fde952f91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.598ex; height:2.676ex;" alt="{\displaystyle X^{\#}}" loading="lazy"> under this order.

A sublinear function is minimal if and only if it is a real <a href="Linear_functional" class="mw-redirect" title="Linear functional">linear functional</a>.<a href="#cite_note-FOOTNOTENariciBeckenstein2011177–220-1">[1]</a>

<div class="mw-heading mw-heading2"><h2 id="Examples_and_sufficient_conditions">Examples and sufficient conditions</h2></div>

Every <a href="Norm_(mathematics)" title="Norm (mathematics)">norm</a>, <a href="Seminorm" title="Seminorm">seminorm</a>, and real linear functional is a sublinear function.

The <a href="Identity_function" title="Identity function">identity function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} \to \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \to \mathbb {R} }</annotation>

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</math><img src="./fadf3927293c3bb66c6bb34668923799af3484b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.97ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} \to \mathbb {R} }" loading="lazy"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X:=\mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle X:=\mathbb {R} }</annotation>

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</math><img src="./90d9c278ab99543cfeec30ab45f03df268799dd1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.403ex; height:2.176ex;" alt="{\displaystyle X:=\mathbb {R} }" loading="lazy"> is an example of a sublinear function (in fact, it is even a linear functional) that is neither positive nor a seminorm; the same is true of this map's negation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto -x.}">

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<annotation encoding="application/x-tex">{\displaystyle x\mapsto -x.}</annotation>

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</math><img src="./031889143fdbe0b9689c8e4446f2e120397dbdfe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.728ex; height:2.176ex;" alt="{\displaystyle x\mapsto -x.}" loading="lazy"><a href="#cite_note-FOOTNOTENariciBeckenstein2011177–221-6">[5]</a>

More generally, for any real <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\leq b,}">

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<annotation encoding="application/x-tex">{\displaystyle a\leq b,}</annotation>

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</math><img src="./fda2fb7363a6f92f8b73a3713f06f1f434ef79f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.973ex; height:2.509ex;" alt="{\displaystyle a\leq b,}" loading="lazy"> the map

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<annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{4}S_{a,b}:\;&&\mathbb {R} &&\;\to \;&\mathbb {R} \\[0.3ex]&&x&&\;\mapsto \;&{\begin{cases}ax&{\text{ if }}x\leq 0\\bx&{\text{ if }}x\geq 0\\\end{cases}}\\\end{alignedat}}}</annotation>

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is a sublinear function on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X:=\mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle X:=\mathbb {R} }</annotation>

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</math><img src="./90d9c278ab99543cfeec30ab45f03df268799dd1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.403ex; height:2.176ex;" alt="{\displaystyle X:=\mathbb {R} }" loading="lazy"> and moreover, every sublinear function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:\mathbb {R} \to \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle p:\mathbb {R} \to \mathbb {R} }</annotation>

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</math><img src="./fbba0b0a051e6ebd646440a50eb872aaad992b47.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.166ex; height:2.509ex;" alt="{\displaystyle p:\mathbb {R} \to \mathbb {R} }" loading="lazy"> is of this form; specifically, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a:=-p(-1)}">

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<annotation encoding="application/x-tex">{\displaystyle a:=-p(-1)}</annotation>

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</math><img src="./9e12c9263fff11f5f2dd708266f5df13d929f96f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.733ex; height:2.843ex;" alt="{\displaystyle a:=-p(-1)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b:=p(1)}">

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<annotation encoding="application/x-tex">{\displaystyle b:=p(1)}</annotation>

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</math><img src="./aec094141b30e694173948e92115cef7bc23eb60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.884ex; height:2.843ex;" alt="{\displaystyle b:=p(1)}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\leq b}">

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<annotation encoding="application/x-tex">{\displaystyle a\leq b}</annotation>

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</math><img src="./41558abc50886fdf38817495b243958d7b3dd39b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.326ex; height:2.343ex;" alt="{\displaystyle a\leq b}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=S_{a,b}.}">

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<annotation encoding="application/x-tex">{\displaystyle p=S_{a,b}.}</annotation>

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</math><img src="./a36d0ecef7472b7f1be8ab18f7f45981b3e4383a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:8.694ex; height:2.843ex;" alt="{\displaystyle p=S_{a,b}.}" loading="lazy">

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

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<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

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</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> are sublinear functions on a real vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> then so is the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto \max\{p(x),q(x)\}.}">

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<annotation encoding="application/x-tex">{\displaystyle x\mapsto \max\{p(x),q(x)\}.}</annotation>

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</math><img src="./e04f307caedf4f019ef4fca548f72774a71daa4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.792ex; height:2.843ex;" alt="{\displaystyle x\mapsto \max\{p(x),q(x)\}.}" loading="lazy"> More generally, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation>

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</math><img src="./10d6ec962de5797ba4f161c40e66dca74ae95cc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}" loading="lazy"> is any non-empty collection of sublinear functionals on a real vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> and if for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X,}">

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<annotation encoding="application/x-tex">{\displaystyle x\in X,}</annotation>

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</math><img src="./e3ebdb0a09f0721ccdd0b779e0a21caf386be82a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.797ex; height:2.509ex;" alt="{\displaystyle x\in X,}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q(x):=\sup\{p(x):p\in {\mathcal {P}}\},}">

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<annotation encoding="application/x-tex">{\displaystyle q(x):=\sup\{p(x):p\in {\mathcal {P}}\},}</annotation>

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</math><img src="./f26f0a6bce6d1fefa66788aa40e3ee0a30eb7ed5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.386ex; height:2.843ex;" alt="{\displaystyle q(x):=\sup\{p(x):p\in {\mathcal {P}}\},}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

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<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

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</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> is a sublinear functional on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}">

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<annotation encoding="application/x-tex">{\displaystyle X.}</annotation>

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</math><img src="./5ba76c5a460c4a0bb1639a193bc1830f0a773e03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" loading="lazy"><a href="#cite_note-FOOTNOTENariciBeckenstein2011177–221-6">[5]</a>

A function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:X\to \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle p:X\to \mathbb {R} }</annotation>

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</math><img src="./c8540db038c97641c58fdf2dbe344cc730d8b441.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.468ex; height:2.509ex;" alt="{\displaystyle p:X\to \mathbb {R} }" loading="lazy"> which is <a href="Subadditivity" title="Subadditivity">subadditive</a>, <a href="Convex_function" title="Convex function">convex</a>, and satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(0)\leq 0}">

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<annotation encoding="application/x-tex">{\displaystyle p(0)\leq 0}</annotation>

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</math><img src="./1c6550e0aab6c9cd03cae158403d37d8719da901.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:8.492ex; height:2.843ex;" alt="{\displaystyle p(0)\leq 0}" loading="lazy"> is also positively homogeneous (the latter condition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(0)\leq 0}">

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<annotation encoding="application/x-tex">{\displaystyle p(0)\leq 0}</annotation>

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</math><img src="./1c6550e0aab6c9cd03cae158403d37d8719da901.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:8.492ex; height:2.843ex;" alt="{\displaystyle p(0)\leq 0}" loading="lazy"> is necessary as the example of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x):={\sqrt {x^{2}+1}}}">

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<annotation encoding="application/x-tex">{\displaystyle p(x):={\sqrt {x^{2}+1}}}</annotation>

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</math><img src="./f26a52bbcc4cdfa3466689ee084de802e4cc8d56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:16.854ex; height:3.509ex;" alt="{\displaystyle p(x):={\sqrt {x^{2}+1}}}" loading="lazy"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X:=\mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle X:=\mathbb {R} }</annotation>

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</math><img src="./90d9c278ab99543cfeec30ab45f03df268799dd1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.403ex; height:2.176ex;" alt="{\displaystyle X:=\mathbb {R} }" loading="lazy"> shows). If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> is positively homogeneous, it is convex if and only if it is subadditive. Therefore, assuming <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(0)\leq 0}">

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<annotation encoding="application/x-tex">{\displaystyle p(0)\leq 0}</annotation>

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</math><img src="./1c6550e0aab6c9cd03cae158403d37d8719da901.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:8.492ex; height:2.843ex;" alt="{\displaystyle p(0)\leq 0}" loading="lazy">, any two properties among subadditivity, convexity, and positive homogeneity implies the third.

<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

Every sublinear function is a <a href="Convex_function" title="Convex function">convex function</a>: For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq t\leq 1,}">

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<annotation encoding="application/x-tex">{\displaystyle 0\leq t\leq 1,}</annotation>

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</math><img src="./d2f346e8c1e6f304824cb6692078da5f000dd9b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.008ex; height:2.509ex;" alt="{\displaystyle 0\leq t\leq 1,}" loading="lazy">

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<mtext> subadditivity</mtext>

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<mtext> nonnegative homogeneity</mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{3}p(tx+(1-t)y)&\leq p(tx)+p((1-t)y)&&\quad {\text{ subadditivity}}\\&=tp(x)+(1-t)p(y)&&\quad {\text{ nonnegative homogeneity}}\\\end{alignedat}}}</annotation>

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If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:X\to \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle p:X\to \mathbb {R} }</annotation>

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</math><img src="./c8540db038c97641c58fdf2dbe344cc730d8b441.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.468ex; height:2.509ex;" alt="{\displaystyle p:X\to \mathbb {R} }" loading="lazy"> is a sublinear function on a vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> then<a href="#cite_note-NullAtZeroAndSumUpperBound-7">[proof 2]</a><a href="#cite_note-FOOTNOTENariciBeckenstein2011120–121-3">[3]</a>

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<annotation encoding="application/x-tex">{\displaystyle p(0)~=~0~\leq ~p(x)+p(-x),}</annotation>

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for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X,}">

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<annotation encoding="application/x-tex">{\displaystyle x\in X,}</annotation>

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</math><img src="./e3ebdb0a09f0721ccdd0b779e0a21caf386be82a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.797ex; height:2.509ex;" alt="{\displaystyle x\in X,}" loading="lazy"> which implies that at least one of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)}">

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<annotation encoding="application/x-tex">{\displaystyle p(x)}</annotation>

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</math><img src="./8cb7afced134ef75572e5314a5d278c2d644f438.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.398ex; height:2.843ex;" alt="{\displaystyle p(x)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(-x)}">

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<annotation encoding="application/x-tex">{\displaystyle p(-x)}</annotation>

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</math><img src="./5d7a0e9a08052f4fd6d6be18b649540b06022429.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:6.206ex; height:2.843ex;" alt="{\displaystyle p(-x)}" loading="lazy"> must be nonnegative; that is, for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X,}">

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<annotation encoding="application/x-tex">{\displaystyle x\in X,}</annotation>

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</math><img src="./e3ebdb0a09f0721ccdd0b779e0a21caf386be82a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.797ex; height:2.509ex;" alt="{\displaystyle x\in X,}" loading="lazy"><a href="#cite_note-FOOTNOTENariciBeckenstein2011120–121-3">[3]</a>

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<annotation encoding="application/x-tex">{\displaystyle 0~\leq ~\max\{p(x),p(-x)\}.}</annotation>

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Moreover, when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:X\to \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle p:X\to \mathbb {R} }</annotation>

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</math><img src="./c8540db038c97641c58fdf2dbe344cc730d8b441.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.468ex; height:2.509ex;" alt="{\displaystyle p:X\to \mathbb {R} }" loading="lazy"> is a sublinear function on a real vector space then the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q:X\to \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle q:X\to \mathbb {R} }</annotation>

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</math><img src="./5282517967c6a6ce35ffdc26ae00b590333dfb12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.279ex; height:2.509ex;" alt="{\displaystyle q:X\to \mathbb {R} }" loading="lazy"> defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q(x)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\max\{p(x),p(-x)\}}">

<mover>

</mrow>

</mrow>

</mstyle>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q(x)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\max\{p(x),p(-x)\}}</annotation>

</semantics>

</math><img src="./c90330d3c4282d0c7b14c28a5f8c91f6f9a0ded4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.675ex; height:3.509ex;" alt="{\displaystyle q(x)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\max\{p(x),p(-x)\}}" loading="lazy"> is a seminorm.<a href="#cite_note-FOOTNOTENariciBeckenstein2011120–121-3">[3]</a>

Subadditivity of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:X\to \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p:X\to \mathbb {R} }</annotation>

</semantics>

</math><img src="./c8540db038c97641c58fdf2dbe344cc730d8b441.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.468ex; height:2.509ex;" alt="{\displaystyle p:X\to \mathbb {R} }" loading="lazy"> guarantees that for all vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in X,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x,y\in X,}</annotation>

</semantics>

</math><img src="./10c8e2c619d7ee4f04b7422de1bf3ac3945c12ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.987ex; height:2.509ex;" alt="{\displaystyle x,y\in X,}" loading="lazy"><a href="#cite_note-FOOTNOTENariciBeckenstein2011177–220-1">[1]</a><a href="#cite_note-ReverseTriangle-8">[proof 3]</a>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(x)-p(y)~\leq ~p(x-y),}</annotation>

</semantics>

</math>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -p(x)~\leq ~p(-x),}</annotation>

</semantics>

</math>

so if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> is also <a href="#symmetric_function">symmetric</a> then the <a href="Reverse_triangle_inequality" class="mw-redirect" title="Reverse triangle inequality">reverse triangle inequality</a> will hold for all vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in X,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x,y\in X,}</annotation>

</semantics>

</math><img src="./10c8e2c619d7ee4f04b7422de1bf3ac3945c12ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.987ex; height:2.509ex;" alt="{\displaystyle x,y\in X,}" loading="lazy">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |p(x)-p(y)|~\leq ~p(x-y).}</annotation>

</semantics>

</math>

Defining <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ker p~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~p^{-1}(0),}">

<mover>

</mrow>

</mrow>

</mstyle>

</mrow>

</mover>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ker p~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~p^{-1}(0),}</annotation>

</semantics>

</math><img src="./baf99bd71bf91a06723a6daefbd3ef5bd6221744.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.818ex; height:3.509ex;" alt="{\displaystyle \ker p~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~p^{-1}(0),}" loading="lazy"> then subadditivity also guarantees that for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in X,}</annotation>

</semantics>

</math><img src="./e3ebdb0a09f0721ccdd0b779e0a21caf386be82a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.797ex; height:2.509ex;" alt="{\displaystyle x\in X,}" loading="lazy"> the value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> on the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+(\ker p\cap -\ker p)=\{x+k:p(k)=0=p(-k)\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x+(\ker p\cap -\ker p)=\{x+k:p(k)=0=p(-k)\}}</annotation>

</semantics>

</math><img src="./785a7ff629b887b5a6350c4c45c7a9565c3a6cb9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.503ex; height:2.843ex;" alt="{\displaystyle x+(\ker p\cap -\ker p)=\{x+k:p(k)=0=p(-k)\}}" loading="lazy"> is constant and equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x).}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(x).}</annotation>

</semantics>

</math><img src="./247c8f81cd0f4a77bffe702ef0b6842884b727a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:5.045ex; height:2.843ex;" alt="{\displaystyle p(x).}" loading="lazy"><a href="#cite_note-ConstantOnEquivClasses-9">[proof 4]</a>

In particular, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ker p=p^{-1}(0)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ker p=p^{-1}(0)}</annotation>

</semantics>

</math><img src="./80333915f20e845faf25b8c460147e3da64b4eae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.3ex; height:3.176ex;" alt="{\displaystyle \ker p=p^{-1}(0)}" loading="lazy"> is a vector subspace of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\ker p=\ker p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -\ker p=\ker p}</annotation>

</semantics>

</math><img src="./bcda899b4419b69bd522958ac034ce70af327073.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.75ex; height:2.509ex;" alt="{\displaystyle -\ker p=\ker p}" loading="lazy"> and the assignment <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+\ker p\mapsto p(x),}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x+\ker p\mapsto p(x),}</annotation>

</semantics>

</math><img src="./da05c88a94732aa6d04b604d30375c8d18d17fb8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.467ex; height:2.843ex;" alt="{\displaystyle x+\ker p\mapsto p(x),}" loading="lazy"> which will be denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {p}},}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {p}},}</annotation>

</semantics>

</math><img src="./d90847bca65036338ad3df1eee270309e1efffb3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.096ex; height:2.509ex;" alt="{\displaystyle {\hat {p}},}" loading="lazy"> is a well-defined real-valued sublinear function on the <a href="Quotient_space_(linear_algebra)" title="Quotient space (linear algebra)">quotient space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\,/\,\ker p}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\,/\,\ker p}</annotation>

</semantics>

</math><img src="./0af298c1a6d459ac18911880670e8b6b33b8123b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.032ex; height:2.843ex;" alt="{\displaystyle X\,/\,\ker p}" loading="lazy"> that satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {p}}^{-1}(0)=\ker p.}">

<msup>

<mover>

</mover>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {p}}^{-1}(0)=\ker p.}</annotation>

</semantics>

</math><img src="./a9fb72d0e8dba849d185350887ee3db148db11da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:15.227ex; height:3.176ex;" alt="{\displaystyle {\hat {p}}^{-1}(0)=\ker p.}" loading="lazy"> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> is a seminorm then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {p}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {p}}}</annotation>

</semantics>

</math><img src="./8bd4c026f1b3413adc58b9b65e89e62bce92c85a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.449ex; height:2.509ex;" alt="{\displaystyle {\hat {p}}}" loading="lazy"> is just the usual canonical norm on the quotient space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\,/\,\ker p.}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\,/\,\ker p.}</annotation>

</semantics>

</math><img src="./6905e21abf592ae73686f3ca0c0a244613c5a755.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.679ex; height:2.843ex;" alt="{\displaystyle X\,/\,\ker p.}" loading="lazy">

/* start https://en.wikipedia.org/ */

.mw-parser-output .math_theorem{margin:1em 2em;padding:0.5em 1em 0.4em;border:1px solid #aaa;overflow:hidden}@media(max-width:500px){.mw-parser-output .math_theorem{margin:1em 0em;padding:0.5em 0.5em 0.4em}}

/* end https://en.wikipedia.org/ */

</style><div class="math_theorem" style="">

Pryce's sublinearity lemma<a href="#cite_note-FOOTNOTESchechter1996313–315-2">[2]</a>—Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:X\to \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p:X\to \mathbb {R} }</annotation>

</semantics>

</math><img src="./c8540db038c97641c58fdf2dbe344cc730d8b441.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.468ex; height:2.509ex;" alt="{\displaystyle p:X\to \mathbb {R} }" loading="lazy"> is a sublinear functional on a vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> and that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\subseteq X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K\subseteq X}</annotation>

</semantics>

</math><img src="./4ba18595612961c1a8de8fbb2464e9bcfd4554da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.144ex; height:2.343ex;" alt="{\displaystyle K\subseteq X}" loading="lazy"> is a non-empty convex subset.

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation>

</semantics>

</math><img src="./3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> is a vector and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,c>0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a,c>0}</annotation>

</semantics>

</math><img src="./a0e078c7a17fdde32457f3898336de706a25f5b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.532ex; height:2.509ex;" alt="{\displaystyle a,c>0}" loading="lazy"> are positive real numbers such that

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(x)+ac~<~\inf _{k\in K}p(x+ak)}</annotation>

</semantics>

</math>

then for every positive real <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b>0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b>0}</annotation>

</semantics>

</math><img src="./94436473a90bd55191a79c59474cb5456dcbec00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.258ex; height:2.176ex;" alt="{\displaystyle b>0}" loading="lazy"> there exists some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {z} \in K}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {z} \in K}</annotation>

</semantics>

</math><img src="./32d4e60dab2b22d7949af99d0888d4b4e294747a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.095ex; height:2.176ex;" alt="{\displaystyle \mathbf {z} \in K}" loading="lazy"> such that

</mrow>

</mrow>

</munder>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(x+a\mathbf {z} )+bc~<~\inf _{k\in K}p(x+a\mathbf {z} +bk).}</annotation>

</semantics>

</math>

</div>

Adding <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle bc}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle bc}</annotation>

</semantics>

</math><img src="./729e20113e9029b3d860aab123277d8aa0d7a950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.004ex; height:2.176ex;" alt="{\displaystyle bc}" loading="lazy"> to both sides of the hypothesis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle p(x)+ac\,<\,\inf _{}p(x+aK)}">

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle p(x)+ac\,<\,\inf _{}p(x+aK)}</annotation>

</semantics>

</math><img src="./aaead24fcb7abb7762b2e9c13cbbebe65e2c6993.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:26.83ex; height:2.843ex;" alt="{\textstyle p(x)+ac\,<\,\inf _{}p(x+aK)}" loading="lazy"> (where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x+aK)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{p(x+ak):k\in K\}}">

<mover>

</mrow>

</mrow>

</mstyle>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(x+aK)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{p(x+ak):k\in K\}}</annotation>

</semantics>

</math><img src="./396f3cb53eb90f578ba8c72800c0c481dff00714.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:33.473ex; height:3.509ex;" alt="{\displaystyle p(x+aK)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{p(x+ak):k\in K\}}" loading="lazy">) and combining that with the conclusion gives

</mrow>

</munder>

</mrow>

</mrow>

</munder>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(x)+ac+bc~<~\inf _{}p(x+aK)+bc~\leq ~p(x+a\mathbf {z} )+bc~<~\inf _{}p(x+a\mathbf {z} +bK)}</annotation>

</semantics>

</math>

which yields many more inequalities, including, for instance,

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(x)+ac+bc~<~p(x+a\mathbf {z} )+bc~<~p(x+a\mathbf {z} +b\mathbf {z} )}</annotation>

</semantics>

</math>

in which an expression on one side of a strict inequality <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,<\,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,<\,}</annotation>

</semantics>

</math><img src="./f5ebb5b330e53c9b9af8e7d7c8e0590d3a5f631e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.582ex; height:1.843ex;" alt="{\displaystyle \,<\,}" loading="lazy"> can be obtained from the other by replacing the symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c}</annotation>

</semantics>

</math><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {z} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {z} }</annotation>

</semantics>

</math><img src="./82eca5d0928078d5a61b9e7e98cc73db31070909.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.188ex; height:1.676ex;" alt="{\displaystyle \mathbf {z} }" loading="lazy"> (or vice versa) and moving the closing parenthesis to the right (or left) of an adjacent summand (all other symbols remain fixed and unchanged).

<div class="mw-heading mw-heading3"><h3 id="Associated_seminorm">Associated seminorm</h3></div>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:X\to \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p:X\to \mathbb {R} }</annotation>

</semantics>

</math><img src="./c8540db038c97641c58fdf2dbe344cc730d8b441.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.468ex; height:2.509ex;" alt="{\displaystyle p:X\to \mathbb {R} }" loading="lazy"> is a real-valued sublinear function on a real vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> (or if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is complex, then when it is considered as a real vector space) then the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q(x)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\max\{p(x),p(-x)\}}">

<mover>

</mrow>

</mrow>

</mstyle>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q(x)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\max\{p(x),p(-x)\}}</annotation>

</semantics>

</math><img src="./c90330d3c4282d0c7b14c28a5f8c91f6f9a0ded4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.675ex; height:3.509ex;" alt="{\displaystyle q(x)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\max\{p(x),p(-x)\}}" loading="lazy"> defines a <a href="Seminorm" title="Seminorm">seminorm</a> on the real vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> called the seminorm associated with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p.}</annotation>

</semantics>

</math><img src="./88532f4eab1d4cef71ef96c0f8c98cac36fd9257.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.906ex; height:2.009ex;" alt="{\displaystyle p.}" loading="lazy"><a href="#cite_note-FOOTNOTENariciBeckenstein2011120–121-3">[3]</a>

A sublinear function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> on a real or complex vector space is a <a href="#symmetric_function">symmetric function</a> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p=q}</annotation>

</semantics>

</math><img src="./4cf52da7809003406aee7e680eb5476a05a5c546.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.427ex; height:2.009ex;" alt="{\displaystyle p=q}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q(x)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\max\{p(x),p(-x)\}}">

<mover>

</mrow>

</mrow>

</mstyle>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q(x)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\max\{p(x),p(-x)\}}</annotation>

</semantics>

</math><img src="./c90330d3c4282d0c7b14c28a5f8c91f6f9a0ded4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.675ex; height:3.509ex;" alt="{\displaystyle q(x)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\max\{p(x),p(-x)\}}" loading="lazy"> as before.

More generally, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:X\to \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p:X\to \mathbb {R} }</annotation>

</semantics>

</math><img src="./c8540db038c97641c58fdf2dbe344cc730d8b441.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.468ex; height:2.509ex;" alt="{\displaystyle p:X\to \mathbb {R} }" loading="lazy"> is a real-valued sublinear function on a (real or complex) vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> then

<mover>

</mrow>

</mrow>

</mstyle>

</mrow>

</mover>

</mrow>

</mrow>

</mrow>

</mrow>

</munder>

<mtext> is a unit scalar </mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q(x)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\sup _{|u|=1}p(ux)~=~\sup\{p(ux):u{\text{ is a unit scalar }}\}}</annotation>

</semantics>

</math>

will define a <a href="Seminorm" title="Seminorm">seminorm</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> if this supremum is always a real number (that is, never equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \infty }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \infty }</annotation>

</semantics>

</math><img src="./c26c105004f30c27aa7c2a9c601550a4183b1f21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }" loading="lazy">).

<div class="mw-heading mw-heading3"><h3 id="Relation_to_linear_functionals">Relation to linear functionals</h3></div>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> is a sublinear function on a real vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> then the following are equivalent:<a href="#cite_note-FOOTNOTENariciBeckenstein2011177–220-1">[1]</a>

<ol>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> is a <a href="Linear_functional" class="mw-redirect" title="Linear functional">linear functional</a>.</li>

<li>for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in X,}</annotation>

</semantics>

</math><img src="./e3ebdb0a09f0721ccdd0b779e0a21caf386be82a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.797ex; height:2.509ex;" alt="{\displaystyle x\in X,}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)+p(-x)\leq 0.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(x)+p(-x)\leq 0.}</annotation>

</semantics>

</math><img src="./b9799e5495c1d28d586e528ece50f6df05d7b046.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:18.263ex; height:2.843ex;" alt="{\displaystyle p(x)+p(-x)\leq 0.}" loading="lazy"></li>

<li>for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in X,}</annotation>

</semantics>

</math><img src="./e3ebdb0a09f0721ccdd0b779e0a21caf386be82a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.797ex; height:2.509ex;" alt="{\displaystyle x\in X,}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)+p(-x)=0.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(x)+p(-x)=0.}</annotation>

</semantics>

</math><img src="./f286058e39c800609daf8bef7eb27ab43d76d9d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:18.263ex; height:2.843ex;" alt="{\displaystyle p(x)+p(-x)=0.}" loading="lazy"></li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> is a minimal sublinear function.</li>

</ol>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> is a sublinear function on a real vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> then there exists a linear functional <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\leq p.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\leq p.}</annotation>

</semantics>

</math><img src="./58005b85054516448df6a8677e2126602d97ed8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.193ex; height:2.509ex;" alt="{\displaystyle f\leq p.}" loading="lazy"><a href="#cite_note-FOOTNOTENariciBeckenstein2011177–220-1">[1]</a>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is a real vector space, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is a linear functional on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X,}</annotation>

</semantics>

</math><img src="./09ba32eeb405f7f5f2bac1eb12987c47d2fd42df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> is a positive sublinear function on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X,}</annotation>

</semantics>

</math><img src="./09ba32eeb405f7f5f2bac1eb12987c47d2fd42df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\leq p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\leq p}</annotation>

</semantics>

</math><img src="./ee9272a49329da37ca0bfb7823b92172188c0d17.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.546ex; height:2.509ex;" alt="{\displaystyle f\leq p}" loading="lazy"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(1)\cap \{x\in X:p(x)<1\}=\varnothing .}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}(1)\cap \{x\in X:p(x)<1\}=\varnothing .}</annotation>

</semantics>

</math><img src="./72c779abeb15838b7802de7efe54e0c17d322424.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.743ex; height:3.176ex;" alt="{\displaystyle f^{-1}(1)\cap \{x\in X:p(x)<1\}=\varnothing .}" loading="lazy"><a href="#cite_note-FOOTNOTENariciBeckenstein2011177–220-1">[1]</a>

<div class="mw-heading mw-heading4"><h4 id="Dominating_a_linear_functional">Dominating a linear functional</h4></div>

A real-valued function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> defined on a subset of a real or complex vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is said to be dominated by a sublinear function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)\leq p(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)\leq p(x)}</annotation>

</semantics>

</math><img src="./c7eaf4021c6a225dfb1c018da4ecfaa6ee669589.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.824ex; height:2.843ex;" alt="{\displaystyle f(x)\leq p(x)}" loading="lazy"> for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> that belongs to the domain of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f.}</annotation>

</semantics>

</math><img src="./ecb3ed2e17fa8f336dcc0fd4b3eddbfb02a50ef3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.925ex; height:2.509ex;" alt="{\displaystyle f.}" loading="lazy">

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:X\to \mathbb {R} }</annotation>

</semantics>

</math><img src="./669fa4832da4b0b229d77eadb270e95188f2eb10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.488ex; height:2.509ex;" alt="{\displaystyle f:X\to \mathbb {R} }" loading="lazy"> is a real <a href="Linear_functional" class="mw-redirect" title="Linear functional">linear functional</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> then<a href="#cite_note-FOOTNOTERudin199156–62-10">[6]</a><a href="#cite_note-FOOTNOTENariciBeckenstein2011177–220-1">[1]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is dominated by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> (that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\leq p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\leq p}</annotation>

</semantics>

</math><img src="./ee9272a49329da37ca0bfb7823b92172188c0d17.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.546ex; height:2.509ex;" alt="{\displaystyle f\leq p}" loading="lazy">) if and only if <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -p(-x)\leq f(x)\leq p(x)\quad {\text{ for every }}x\in X.}">

<mtext> for every </mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -p(-x)\leq f(x)\leq p(x)\quad {\text{ for every }}x\in X.}</annotation>

</semantics>

</math>

Moreover, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> is a seminorm or some other symmetric map (which by definition means that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(-x)=p(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(-x)=p(x)}</annotation>

</semantics>

</math><img src="./6e25594f5a6ea3ae62f0545ad97ff30419766102.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:13.613ex; height:2.843ex;" alt="{\displaystyle p(-x)=p(x)}" loading="lazy"> holds for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">) then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\leq p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\leq p}</annotation>

</semantics>

</math><img src="./ee9272a49329da37ca0bfb7823b92172188c0d17.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.546ex; height:2.509ex;" alt="{\displaystyle f\leq p}" loading="lazy"> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f|\leq p.}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |f|\leq p.}</annotation>

</semantics>

</math><img src="./a2105e8a75fd69f6d72ca95161f24522d1a0388a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.487ex; height:2.843ex;" alt="{\displaystyle |f|\leq p.}" loading="lazy">

Theorem<a href="#cite_note-FOOTNOTENariciBeckenstein2011177–220-1">[1]</a>—If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:X\to \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p:X\to \mathbb {R} }</annotation>

</semantics>

</math><img src="./c8540db038c97641c58fdf2dbe344cc730d8b441.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.468ex; height:2.509ex;" alt="{\displaystyle p:X\to \mathbb {R} }" loading="lazy"> be a sublinear function on a real vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> and if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z\in X}</annotation>

</semantics>

</math><img src="./dff47a41328065a0e15e3a7089cfd392ee954e36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.909ex; height:2.176ex;" alt="{\displaystyle z\in X}" loading="lazy"> then there exists a linear functional <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> that is dominated by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> (that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\leq p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\leq p}</annotation>

</semantics>

</math><img src="./ee9272a49329da37ca0bfb7823b92172188c0d17.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.546ex; height:2.509ex;" alt="{\displaystyle f\leq p}" loading="lazy">) and satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)=p(z).}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(z)=p(z).}</annotation>

</semantics>

</math><img src="./802b29fdaf1080fdc4c04503978e08656c5b14da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.988ex; height:2.843ex;" alt="{\displaystyle f(z)=p(z).}" loading="lazy">

Moreover, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is a <a href="Topological_vector_space" title="Topological vector space">topological vector space</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> is continuous at the origin then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is continuous.

</div>

<div class="mw-heading mw-heading3"><h3 id="Continuity">Continuity</h3></div>

Theorem<a href="#cite_note-FOOTNOTENariciBeckenstein2011192–193-11">[7]</a>—Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:X\to \mathbb {R} }</annotation>

</semantics>

</math><img src="./669fa4832da4b0b229d77eadb270e95188f2eb10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.488ex; height:2.509ex;" alt="{\displaystyle f:X\to \mathbb {R} }" loading="lazy"> is a subadditive function (that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x+y)\leq f(x)+f(y)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x+y)\leq f(x)+f(y)}</annotation>

</semantics>

</math><img src="./c0440ac144d656f8aa4edb50a971376aad15ff3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.013ex; height:2.843ex;" alt="{\displaystyle f(x+y)\leq f(x)+f(y)}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x,y\in X}</annotation>

</semantics>

</math><img src="./6d72f66ab332ed430aa9b34ff18c9723c4fea2a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.34ex; height:2.509ex;" alt="{\displaystyle x,y\in X}" loading="lazy">).

Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is continuous at the origin if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is uniformly continuous on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X.}</annotation>

</semantics>

</math><img src="./5ba76c5a460c4a0bb1639a193bc1830f0a773e03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" loading="lazy">

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(0)=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(0)=0}</annotation>

</semantics>

</math><img src="./8d308c32c9894b88115262081194321ae7d9bbf3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.511ex; height:2.843ex;" alt="{\displaystyle f(0)=0}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is continuous if and only if its absolute value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f|:X\to [0,\infty )}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |f|:X\to [0,\infty )}</annotation>

</semantics>

</math><img src="./05cc3902185c2a78681a50797729991f31d9509a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.175ex; height:2.843ex;" alt="{\displaystyle |f|:X\to [0,\infty )}" loading="lazy"> is continuous.

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is non-negative then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is continuous if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x\in X:f(x)<1\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{x\in X:f(x)<1\}}</annotation>

</semantics>

</math><img src="./629989fcc399a62bcd03e501636d4c474bd25720.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.091ex; height:2.843ex;" alt="{\displaystyle \{x\in X:f(x)<1\}}" loading="lazy"> is open in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X.}</annotation>

</semantics>

</math><img src="./5ba76c5a460c4a0bb1639a193bc1830f0a773e03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" loading="lazy">

</div>

Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is a <a href="Topological_vector_space" title="Topological vector space">topological vector space</a> (TVS) over the real or complex numbers and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> is a sublinear function on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X.}</annotation>

</semantics>

</math><img src="./5ba76c5a460c4a0bb1639a193bc1830f0a773e03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" loading="lazy">

Then the following are equivalent:<a href="#cite_note-FOOTNOTENariciBeckenstein2011192–193-11">[7]</a>

<ol>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> is continuous;</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> is continuous at 0;</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> is uniformly continuous on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">;</li>

</ol>

and if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> is positive then this list may be extended to include:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{x\in X:p(x)<1\}}</annotation>

</semantics>

</math><img src="./0e8864ce6233078002446ee9a743d11063a113e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.982ex; height:2.843ex;" alt="{\displaystyle \{x\in X:p(x)<1\}}" loading="lazy"> is open in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X.}</annotation>

</semantics>

</math><img src="./5ba76c5a460c4a0bb1639a193bc1830f0a773e03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" loading="lazy"></li>

</ol>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is a real TVS, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is a linear functional on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X,}</annotation>

</semantics>

</math><img src="./09ba32eeb405f7f5f2bac1eb12987c47d2fd42df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> is a continuous sublinear function on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X,}</annotation>

</semantics>

</math><img src="./09ba32eeb405f7f5f2bac1eb12987c47d2fd42df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\leq p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\leq p}</annotation>

</semantics>

</math><img src="./ee9272a49329da37ca0bfb7823b92172188c0d17.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.546ex; height:2.509ex;" alt="{\displaystyle f\leq p}" loading="lazy"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> implies that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is continuous.<a href="#cite_note-FOOTNOTENariciBeckenstein2011192–193-11">[7]</a>

<div class="mw-heading mw-heading3"><h3 id="Relation_to_Minkowski_functions_and_open_convex_sets">Relation to Minkowski functions and open convex sets</h3></div>

Theorem<a href="#cite_note-FOOTNOTENariciBeckenstein2011192–193-11">[7]</a>—If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> is a convex open neighborhood of the origin in a <a href="Topological_vector_space" title="Topological vector space">topological vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> then the <a href="Minkowski_functional" title="Minkowski functional">Minkowski functional</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U,}</annotation>

</semantics>

</math><img src="./59c21c569e498e0197798e3428b2bc6f25c0a457.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.429ex; height:2.509ex;" alt="{\displaystyle U,}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{U}:X\to [0,\infty ),}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{U}:X\to [0,\infty ),}</annotation>

</semantics>

</math><img src="./6ee787ead72d32072f7f91ffe862e1e7239ac4bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:17.001ex; height:2.843ex;" alt="{\displaystyle p_{U}:X\to [0,\infty ),}" loading="lazy"> is a continuous non-negative sublinear function on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=\left\{x\in X:p_{U}(x)<1\right\};}">

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U=\left\{x\in X:p_{U}(x)<1\right\};}</annotation>

</semantics>

</math><img src="./0867b8ee8e9359ef3158d7a9d7e66adb2c3bfa8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.389ex; height:2.843ex;" alt="{\displaystyle U=\left\{x\in X:p_{U}(x)<1\right\};}" loading="lazy"> if in addition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> is a <a href="Balanced_set" title="Balanced set">balanced set</a> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{U}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{U}}</annotation>

</semantics>

</math><img src="./3aed538a9d66631c371449bd826251f8f29d57d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.752ex; height:2.009ex;" alt="{\displaystyle p_{U}}" loading="lazy"> is a <a href="Seminorm" title="Seminorm">seminorm</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X.}</annotation>

</semantics>

</math><img src="./5ba76c5a460c4a0bb1639a193bc1830f0a773e03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" loading="lazy">

</div>

<div class="mw-heading mw-heading4"><h4 id="Relation_to_open_convex_sets">Relation to open convex sets</h4></div>

Theorem<a href="#cite_note-FOOTNOTENariciBeckenstein2011192–193-11">[7]</a>—Suppose that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is a <a href="Topological_vector_space" title="Topological vector space">topological vector space</a> (not necessarily <a href="Locally_convex_topological_vector_space" title="Locally convex topological vector space">locally convex</a> or <a href="Hausdorff_space" title="Hausdorff space">Hausdorff</a>) over the real or complex numbers.

Then the open convex subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> are exactly those that are of the form <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z+\{x\in X:p(x)<1\}=\{x\in X:p(x-z)<1\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z+\{x\in X:p(x)<1\}=\{x\in X:p(x-z)<1\}}</annotation>

</semantics>

</math> for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z\in X}</annotation>

</semantics>

</math><img src="./dff47a41328065a0e15e3a7089cfd392ee954e36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.909ex; height:2.176ex;" alt="{\displaystyle z\in X}" loading="lazy"> and some positive continuous sublinear function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X.}</annotation>

</semantics>

</math><img src="./5ba76c5a460c4a0bb1639a193bc1830f0a773e03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" loading="lazy">

</div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .math_proof{border:thin solid #aaa;margin:1em 2em;padding:0.5em 1em 0.4em}@media(max-width:500px){.mw-parser-output .math_proof{margin:1em 0;padding:0.5em 0.5em 0.4em}}

/* end https://en.wikipedia.org/ */

</style><div class="math_proof" style="">Proof

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> be an open convex subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X.}</annotation>

</semantics>

</math><img src="./5ba76c5a460c4a0bb1639a193bc1830f0a773e03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" loading="lazy">

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\in V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0\in V}</annotation>

</semantics>

</math><img src="./6d2f0cd30055c16c302a901c4f0fb6583b0faf89.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.79ex; height:2.176ex;" alt="{\displaystyle 0\in V}" loading="lazy"> then let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z:=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z:=0}</annotation>

</semantics>

</math><img src="./06768fdb16f657da598939cf3acc3e7e2b2d4751.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.996ex; height:2.176ex;" alt="{\displaystyle z:=0}" loading="lazy"> and otherwise let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\in V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z\in V}</annotation>

</semantics>

</math><img src="./b4e530fb380ba488705c57df90a2631e9055b7e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.716ex; height:2.176ex;" alt="{\displaystyle z\in V}" loading="lazy"> be arbitrary.

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:X\to [0,\infty )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p:X\to [0,\infty )}</annotation>

</semantics>

</math><img src="./1ef6cb5abd26f4d5c8096289aca0be38b0ce96f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:14.862ex; height:2.843ex;" alt="{\displaystyle p:X\to [0,\infty )}" loading="lazy"> be the <a href="Minkowski_functional" title="Minkowski functional">Minkowski functional</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V-z,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V-z,}</annotation>

</semantics>

</math><img src="./76ac12d1d9f0b8b8a54b8244bc79546bd18ce1bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.363ex; height:2.509ex;" alt="{\displaystyle V-z,}" loading="lazy"> which is a continuous sublinear function on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V-z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V-z}</annotation>

</semantics>

</math><img src="./d49a7aeadff801a8e764c34c9c78986f753ea458.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.716ex; height:2.343ex;" alt="{\displaystyle V-z}" loading="lazy"> is convex, <a href="Absorbing_set" title="Absorbing set">absorbing</a>, and open (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> however is not necessarily a seminorm since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> was not assumed to be <a href="Balanced_set" title="Balanced set">balanced</a>).

From <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=X-z,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X=X-z,}</annotation>

</semantics>

</math><img src="./42a50415f91c7a4fd25788aa0abf8f81ae4f7869.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.634ex; height:2.509ex;" alt="{\displaystyle X=X-z,}" loading="lazy"> it follows that

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z+\{x\in X:p(x)<1\}=\{x\in X:p(x-z)<1\}.}</annotation>

</semantics>

</math>

It will be shown that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=z+\{x\in X:p(x)<1\},}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V=z+\{x\in X:p(x)<1\},}</annotation>

</semantics>

</math><img src="./cae7ff5dd8971989d4e44d01b8611e3d9334d437.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.443ex; height:2.843ex;" alt="{\displaystyle V=z+\{x\in X:p(x)<1\},}" loading="lazy"> which will complete the proof.

One of the known <a href="Minkowski_functional#Properties" title="Minkowski functional">properties of Minkowski functionals</a> guarantees <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \{x\in X:p(x)<1\}=(0,1)(V-z),}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \{x\in X:p(x)<1\}=(0,1)(V-z),}</annotation>

</semantics>

</math><img src="./d2e14ba6b44f8c22acf6de479c51ebd63decc5e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.42ex; height:2.843ex;" alt="{\textstyle \{x\in X:p(x)<1\}=(0,1)(V-z),}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,1)(V-z)\;{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\;\{tx:0<t<1,x\in V-z\}=V-z}">

<mover>

</mrow>

</mrow>

</mstyle>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (0,1)(V-z)\;{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\;\{tx:0<t<1,x\in V-z\}=V-z}</annotation>

</semantics>

</math><img src="./ce77c101fa3a36679425c76ef0da5ebbe7343a58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:51.319ex; height:3.509ex;" alt="{\displaystyle (0,1)(V-z)\;{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\;\{tx:0<t<1,x\in V-z\}=V-z}" loading="lazy"> since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V-z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V-z}</annotation>

</semantics>

</math><img src="./d49a7aeadff801a8e764c34c9c78986f753ea458.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.716ex; height:2.343ex;" alt="{\displaystyle V-z}" loading="lazy"> is convex and contains the origin.

Thus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V-z=\{x\in X:p(x)<1\},}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V-z=\{x\in X:p(x)<1\},}</annotation>

</semantics>

</math><img src="./c90e936e59231373a4d3fb3f4e507871799ac266.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.443ex; height:2.843ex;" alt="{\displaystyle V-z=\{x\in X:p(x)<1\},}" loading="lazy"> as desired. <a href="Q.E.D." title="Q.E.D."><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \blacksquare }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \blacksquare }</annotation>

</semantics>

</math><img src="./8733090f2d787d03101c3e16dc3f6404f0e7dd4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \blacksquare }" loading="lazy"></a>

</div>

<div class="mw-heading mw-heading2"><h2 id="Operators">Operators</h2></div>

The concept can be extended to operators that are homogeneous and subadditive.

This requires only that the <a href="Codomain" title="Codomain">codomain</a> be, say, an <a href="Ordered_vector_space" title="Ordered vector space">ordered vector space</a> to make sense of the conditions.

<div class="mw-heading mw-heading2"><h2 id="Computer_science_definition">Computer science definition</h2></div>

In <a href="Computer_science" title="Computer science">computer science</a>, a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {Z} ^{+}\to \mathbb {R} }">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:\mathbb {Z} ^{+}\to \mathbb {R} }</annotation>

</semantics>

</math><img src="./63d25f6a09065b87a64766c88e010426c109fa63.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.569ex; height:2.843ex;" alt="{\displaystyle f:\mathbb {Z} ^{+}\to \mathbb {R} }" loading="lazy"> is called sublinear if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }{\frac {f(n)}{n}}=0,}">

</mrow>

</munder>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }{\frac {f(n)}{n}}=0,}</annotation>

</semantics>

</math><img src="./1932a712238a41c0752a48d1a29efdb8b6447510.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.886ex; height:5.676ex;" alt="{\displaystyle \lim _{n\to \infty }{\frac {f(n)}{n}}=0,}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)\in o(n)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(n)\in o(n)}</annotation>

</semantics>

</math><img src="./d5e2696675583144850e7cba13c33c794e208fe6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.655ex; height:2.843ex;" alt="{\displaystyle f(n)\in o(n)}" loading="lazy"> in <a href="Big_O_notation#Little-o_notation" title="Big O notation">asymptotic notation</a> (notice the small <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle o}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle o}</annotation>

</semantics>

</math><img src="./0c1031f61947aa3d1cf3a70ec3e4904df2c3675d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle o}" loading="lazy">).

Formally, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)\in o(n)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(n)\in o(n)}</annotation>

</semantics>

</math><img src="./d5e2696675583144850e7cba13c33c794e208fe6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.655ex; height:2.843ex;" alt="{\displaystyle f(n)\in o(n)}" loading="lazy"> if and only if, for any given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c>0,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c>0,}</annotation>

</semantics>

</math><img src="./53df663fbf9022d4aa6cfe5489aa11fd0af3992d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.915ex; height:2.509ex;" alt="{\displaystyle c>0,}" loading="lazy"> there exists an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)<cn}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(n)<cn}</annotation>

</semantics>

</math><img src="./835a78e48bfc06e113acc092c741d9418e7f4ead.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.983ex; height:2.843ex;" alt="{\displaystyle f(n)<cn}" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq N.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\geq N.}</annotation>

</semantics>

</math><img src="./ec9a5566b18ef2b412f4a14c159ac959d126f603.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.204ex; height:2.343ex;" alt="{\displaystyle n\geq N.}" loading="lazy"><a href="#cite_note-12">[8]</a>

That is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> grows slower than any linear function.

The two meanings should not be confused: while a Banach functional is <a href="Convex_function" title="Convex function">convex</a>, almost the opposite is true for functions of sublinear growth: every function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)\in o(n)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(n)\in o(n)}</annotation>

</semantics>

</math><img src="./d5e2696675583144850e7cba13c33c794e208fe6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.655ex; height:2.843ex;" alt="{\displaystyle f(n)\in o(n)}" loading="lazy"> can be upper-bounded by a <a href="Concave_function" title="Concave function">concave function</a> of sublinear growth.<a href="#cite_note-13">[9]</a>

<ul><li><a href="Asymmetric_norm" title="Asymmetric norm">Asymmetric norm</a> – Generalization of the concept of a norm</li>

<li><a href="Auxiliary_normed_space" title="Auxiliary normed space">Auxiliary normed space</a></li>

<li><a href="Hahn-Banach_theorem" class="mw-redirect" title="Hahn-Banach theorem">Hahn-Banach theorem</a> – Theorem on extension of bounded linear functionalsPages displaying short descriptions of redirect targets</li>

<li><a href="Linear_functional" class="mw-redirect" title="Linear functional">Linear functional</a> – Linear map from a vector space to its field of scalarsPages displaying short descriptions of redirect targets</li>

<li><a href="Minkowski_functional" title="Minkowski functional">Minkowski functional</a> – Function made from a set</li>

<li><a href="Norm_(mathematics)" title="Norm (mathematics)">Norm (mathematics)</a> – Length in a vector space</li>

<li><a href="Seminorm" title="Seminorm">Seminorm</a> – Mathematical function</li>

<li><a href="Superadditivity" title="Superadditivity">Superadditivity</a> – Property of a function</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

/* start https://en.wikipedia.org/ */

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</style><div class="reflist">

</div>

Proofs

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in X.}</annotation>

</semantics>

</math><img src="./0deab6a01578b5b543b772df12dc0d2c593cc924.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.797ex; height:2.176ex;" alt="{\displaystyle x\in X.}" loading="lazy"> The triangle inequality and symmetry imply <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(0)=p(x+(-x))\leq p(x)+p(-x)=p(x)+p(x)=2p(x).}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(0)=p(x+(-x))\leq p(x)+p(-x)=p(x)+p(x)=2p(x).}</annotation>

</semantics>

</math><img src="./600b8cc115cf3895dc859e91991a23face6e7c52.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:59.561ex; height:2.843ex;" alt="{\displaystyle p(0)=p(x+(-x))\leq p(x)+p(-x)=p(x)+p(x)=2p(x).}" loading="lazy"> Substituting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

</semantics>

</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> and then subtracting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(0)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(0)}</annotation>

</semantics>

</math><img src="./f6d076cd2e950ff98d225d229dae5bd287aab105.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.231ex; height:2.843ex;" alt="{\displaystyle p(0)}" loading="lazy"> from both sides proves that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq p(0).}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0\leq p(0).}</annotation>

</semantics>

</math><img src="./66f2ea851c12abc6ba53f81645565aa05949457f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.049ex; height:2.843ex;" alt="{\displaystyle 0\leq p(0).}" loading="lazy"> Thus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq p(0)\leq 2p(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0\leq p(0)\leq 2p(x)}</annotation>

</semantics>

</math><img src="./e38359a507cdca71cfe6c91ca845a28af691f4a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.971ex; height:2.843ex;" alt="{\displaystyle 0\leq p(0)\leq 2p(x)}" loading="lazy"> which implies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq p(x).}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0\leq p(x).}</annotation>

</semantics>

</math><img src="./250d7b3bbdb2060f64d3d2ddbbc272bfabf6e3b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.216ex; height:2.843ex;" alt="{\displaystyle 0\leq p(x).}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \blacksquare }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \blacksquare }</annotation>

</semantics>

</math><img src="./8733090f2d787d03101c3e16dc3f6404f0e7dd4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \blacksquare }" loading="lazy">

</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation>

</semantics>

</math><img src="./3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r:=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r:=0}</annotation>

</semantics>

</math><img src="./5981c187b517310390607bf77ad352e4be298f8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.956ex; height:2.176ex;" alt="{\displaystyle r:=0}" loading="lazy"> then nonnegative homogeneity implies that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(0)=p(rx)=rp(x)=0p(x)=0.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(0)=p(rx)=rp(x)=0p(x)=0.}</annotation>

</semantics>

</math><img src="./734bb785e72f239ba58a86ff682eaa02b887ab94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:34.619ex; height:2.843ex;" alt="{\displaystyle p(0)=p(rx)=rp(x)=0p(x)=0.}" loading="lazy"> Consequently, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=p(0)=p(x+(-x))\leq p(x)+p(-x),}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0=p(0)=p(x+(-x))\leq p(x)+p(-x),}</annotation>

</semantics>

</math><img src="./692268eadb3669df0fdf3af459cd08c610c68212.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.607ex; height:2.843ex;" alt="{\displaystyle 0=p(0)=p(x+(-x))\leq p(x)+p(-x),}" loading="lazy"> which is only possible if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq \max\{p(x),p(-x)\}.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0\leq \max\{p(x),p(-x)\}.}</annotation>

</semantics>

</math><img src="./6a9ea75f44bab44eb6bb143c63c0e3a484dbf89c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.017ex; height:2.843ex;" alt="{\displaystyle 0\leq \max\{p(x),p(-x)\}.}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \blacksquare }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \blacksquare }</annotation>

</semantics>

</math><img src="./8733090f2d787d03101c3e16dc3f6404f0e7dd4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \blacksquare }" loading="lazy">

</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(x)=p(y+(x-y))\leq p(y)+p(x-y),}</annotation>

</semantics>

</math><img src="./23de721391b80d0e6ec915489cfbeef81fe91a43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:40.63ex; height:2.843ex;" alt="{\displaystyle p(x)=p(y+(x-y))\leq p(y)+p(x-y),}" loading="lazy"> which happens if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)-p(y)\leq p(x-y).}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(x)-p(y)\leq p(x-y).}</annotation>

</semantics>

</math><img src="./5bb4383855d7c98bd91f231df65556daf2886a55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:23.422ex; height:2.843ex;" alt="{\displaystyle p(x)-p(y)\leq p(x-y).}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \blacksquare }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \blacksquare }</annotation>

</semantics>

</math><img src="./8733090f2d787d03101c3e16dc3f6404f0e7dd4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \blacksquare }" loading="lazy"> Substituting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y:=-x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y:=-x}</annotation>

</semantics>

</math><img src="./43c1920ac1e09b851239e9d07c0b7f1d3e11ca45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.039ex; height:2.343ex;" alt="{\displaystyle y:=-x}" loading="lazy"> and gives <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)-p(-x)\leq p(x-(-x))=p(x+x)\leq p(x)+p(x),}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(x)-p(-x)\leq p(x-(-x))=p(x+x)\leq p(x)+p(x),}</annotation>

</semantics>

</math><img src="./0d8cc2724ab738889cbcbc63218386304e738a5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:55.329ex; height:2.843ex;" alt="{\displaystyle p(x)-p(-x)\leq p(x-(-x))=p(x+x)\leq p(x)+p(x),}" loading="lazy"> which implies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -p(-x)\leq p(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -p(-x)\leq p(x)}</annotation>

</semantics>

</math><img src="./81ef19f56a34c28c728a2eff65cbed653a6e1a67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.332ex; height:2.843ex;" alt="{\displaystyle -p(-x)\leq p(x)}" loading="lazy"> (positive homogeneity is not needed; the triangle inequality suffices). <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \blacksquare }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \blacksquare }</annotation>

</semantics>

</math><img src="./8733090f2d787d03101c3e16dc3f6404f0e7dd4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \blacksquare }" loading="lazy">

</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation>

</semantics>

</math><img src="./3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in p^{-1}(0)\cap (-p^{-1}(0)).}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k\in p^{-1}(0)\cap (-p^{-1}(0)).}</annotation>

</semantics>

</math><img src="./5862796ff497493262982533d8d03e08e86fe881.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.847ex; height:3.176ex;" alt="{\displaystyle k\in p^{-1}(0)\cap (-p^{-1}(0)).}" loading="lazy"> It remains to show that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x+k)=p(x).}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(x+k)=p(x).}</annotation>

</semantics>

</math><img src="./c4d13542c3227c1e32833e84e98323ba97912bb0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:16.503ex; height:2.843ex;" alt="{\displaystyle p(x+k)=p(x).}" loading="lazy"> The triangle inequality implies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x+k)\leq p(x)+p(k)=p(x)+0=p(x).}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(x+k)\leq p(x)+p(k)=p(x)+0=p(x).}</annotation>

</semantics>

</math><img src="./58fb3993c332da2d2f240241adc0bdcd71c7af16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:42.35ex; height:2.843ex;" alt="{\displaystyle p(x+k)\leq p(x)+p(k)=p(x)+0=p(x).}" loading="lazy"> Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(-k)=0,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(-k)=0,}</annotation>

</semantics>

</math><img src="./9700457cca831f6702ebc38b43dde0eaa57b02bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:10.995ex; height:2.843ex;" alt="{\displaystyle p(-k)=0,}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)=p(x)-p(-k)\leq p(x-(-k))=p(x+k),}">

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<annotation encoding="application/x-tex">{\displaystyle p(x)=p(x)-p(-k)\leq p(x-(-k))=p(x+k),}</annotation>

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</math><img src="./0ec3347619eec7694ba69707624215fc5f9afd03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:47.824ex; height:2.843ex;" alt="{\displaystyle p(x)=p(x)-p(-k)\leq p(x-(-k))=p(x+k),}" loading="lazy"> as desired. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \blacksquare }">

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<annotation encoding="application/x-tex">{\displaystyle \blacksquare }</annotation>

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</math><img src="./8733090f2d787d03101c3e16dc3f6404f0e7dd4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \blacksquare }" loading="lazy">

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<li id="cite_note-FOOTNOTENariciBeckenstein2011177–220-1">^ <a href="#cite_ref-FOOTNOTENariciBeckenstein2011177–220_1-0">a</a> <a href="#cite_ref-FOOTNOTENariciBeckenstein2011177–220_1-1">b</a> <a href="#cite_ref-FOOTNOTENariciBeckenstein2011177–220_1-2">c</a> <a href="#cite_ref-FOOTNOTENariciBeckenstein2011177–220_1-3">d</a> <a href="#cite_ref-FOOTNOTENariciBeckenstein2011177–220_1-4">e</a> <a href="#cite_ref-FOOTNOTENariciBeckenstein2011177–220_1-5">f</a> <a href="#cite_ref-FOOTNOTENariciBeckenstein2011177–220_1-6">g</a> <a href="#cite_ref-FOOTNOTENariciBeckenstein2011177–220_1-7">h</a> <a href="#cite_ref-FOOTNOTENariciBeckenstein2011177–220_1-8">i</a> <a href="#CITEREFNariciBeckenstein2011">Narici & Beckenstein 2011</a>, pp. 177–220.

</li>

<li id="cite_note-FOOTNOTESchechter1996313–315-2">^ <a href="#cite_ref-FOOTNOTESchechter1996313–315_2-0">a</a> <a href="#cite_ref-FOOTNOTESchechter1996313–315_2-1">b</a> <a href="#cite_ref-FOOTNOTESchechter1996313–315_2-2">c</a> <a href="#CITEREFSchechter1996">Schechter 1996</a>, pp. 313–315.

</li>

<li id="cite_note-FOOTNOTENariciBeckenstein2011120–121-3">^ <a href="#cite_ref-FOOTNOTENariciBeckenstein2011120–121_3-0">a</a> <a href="#cite_ref-FOOTNOTENariciBeckenstein2011120–121_3-1">b</a> <a href="#cite_ref-FOOTNOTENariciBeckenstein2011120–121_3-2">c</a> <a href="#cite_ref-FOOTNOTENariciBeckenstein2011120–121_3-3">d</a> <a href="#cite_ref-FOOTNOTENariciBeckenstein2011120–121_3-4">e</a> <a href="#CITEREFNariciBeckenstein2011">Narici & Beckenstein 2011</a>, pp. 120–121.

</li>

<li id="cite_note-FOOTNOTEKubrusly2011200-4"><a href="#cite_ref-FOOTNOTEKubrusly2011200_4-0">^</a> <a href="#CITEREFKubrusly2011">Kubrusly 2011</a>, p. 200.

</li>

<li id="cite_note-FOOTNOTENariciBeckenstein2011177–221-6">^ <a href="#cite_ref-FOOTNOTENariciBeckenstein2011177–221_6-0">a</a> <a href="#cite_ref-FOOTNOTENariciBeckenstein2011177–221_6-1">b</a> <a href="#CITEREFNariciBeckenstein2011">Narici & Beckenstein 2011</a>, pp. 177–221.

</li>

<li id="cite_note-FOOTNOTERudin199156–62-10"><a href="#cite_ref-FOOTNOTERudin199156–62_10-0">^</a> <a href="#CITEREFRudin1991">Rudin 1991</a>, pp. 56–62.

</li>

<li id="cite_note-FOOTNOTENariciBeckenstein2011192–193-11">^ <a href="#cite_ref-FOOTNOTENariciBeckenstein2011192–193_11-0">a</a> <a href="#cite_ref-FOOTNOTENariciBeckenstein2011192–193_11-1">b</a> <a href="#cite_ref-FOOTNOTENariciBeckenstein2011192–193_11-2">c</a> <a href="#cite_ref-FOOTNOTENariciBeckenstein2011192–193_11-3">d</a> <a href="#cite_ref-FOOTNOTENariciBeckenstein2011192–193_11-4">e</a> <a href="#CITEREFNariciBeckenstein2011">Narici & Beckenstein 2011</a>, pp. 192–193.

</li>

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</style><cite id="CITEREFThomas_H._Cormen,_Charles_E._Leiserson,_Ronald_L._Rivest,_and_Clifford_Stein2001" class="citation book cs1"><a href="Thomas_H._Cormen" title="Thomas H. Cormen">Thomas H. Cormen</a>, <a href="Charles_E._Leiserson" title="Charles E. Leiserson">Charles E. Leiserson</a>, <a href="Ronald_L._Rivest" class="mw-redirect" title="Ronald L. Rivest">Ronald L. Rivest</a>, and <a href="Clifford_Stein" title="Clifford Stein">Clifford Stein</a> (2001) [1990]. "3.1". <a href="Introduction_to_Algorithms" title="Introduction to Algorithms">Introduction to Algorithms</a> (2nd ed.). MIT Press and McGraw-Hill. pp. 47–48. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-262-03293-7</bdi>.</cite><code class="cs1-code">{{cite book}}</code>: CS1 maint: multiple names: authors list (link)

</li>

<li id="cite_note-13"><a href="#cite_ref-13">^</a> <cite id="CITEREFCeccherini-SilbersteinSalvatoriSava-Huss2017" class="citation book cs1">Ceccherini-Silberstein, Tullio; Salvatori, Maura; Sava-Huss, Ecaterina (2017-06-29). Groups, graphs, and random walks. Cambridge. Lemma 5.17. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781316604403</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/948670194">948670194</a>.</cite><code class="cs1-code">{{cite book}}</code>: CS1 maint: location missing publisher (link)

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2></div>

<ul><li><cite id="CITEREFKubrusly2011" class="citation book cs1">Kubrusly, Carlos S. (2011). The Elements of Operator Theory (Second ed.). Boston: <a href="Birkh%C3%A4user" title="Birkhäuser">Birkhäuser</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8176-4998-2</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/710154895">710154895</a>.</cite></li>

<li><cite id="CITEREFRudin1991" class="citation book cs1"><a href="Walter_Rudin" title="Walter Rudin">Rudin, Walter</a> (1991). <a rel="nofollow" class="external text" href="https://archive.org/details/functionalanalys00rudi">Functional Analysis</a>. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: <a href="McGraw-Hill_Science/Engineering/Math" class="mw-redirect" title="McGraw-Hill Science/Engineering/Math">McGraw-Hill Science/Engineering/Math</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-07-054236-5</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/21163277">21163277</a>.</cite></li>

<li><cite id="CITEREFNariciBeckenstein2011" class="citation book cs1">Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1584888666</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/144216834">144216834</a>.</cite></li>

<li><cite id="CITEREFSchaeferWolff1999" class="citation book cs1"><a href="Helmut_H._Schaefer" title="Helmut H. Schaefer">Schaefer, Helmut H.</a>; Wolff, Manfred P. (1999). Topological Vector Spaces. <a href="Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">GTM</a>. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4612-7155-0</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/840278135">840278135</a>.</cite></li>

<li><cite id="CITEREFSchechter1996" class="citation book cs1"><a href="Eric_Schechter" title="Eric Schechter">Schechter, Eric</a> (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-12-622760-4</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/175294365">175294365</a>.</cite></li>

<li><cite id="CITEREFTrèves2006" class="citation book cs1"><a href="Fran%C3%A7ois_Tr%C3%A8ves" title="François Trèves">Trèves, François</a> (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-45352-1</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/853623322">853623322</a>.</cite></li></ul>

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</style><div id="Functional_analysis_(topics_–_glossary)364" style="font-size:114%;margin:0 4em"><a href="Functional_analysis" title="Functional analysis">Functional analysis</a> (<a href="List_of_functional_analysis_topics" title="List of functional analysis topics">topics</a> – <a href="Glossary_of_functional_analysis" title="Glossary of functional analysis">glossary</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Spaces</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Banach_space" title="Banach space">Banach</a></li>

<li><a href="Besov_space" title="Besov space">Besov</a></li>

<li><a href="Fr%C3%A9chet_space" title="Fréchet space">Fréchet</a></li>

<li><a href="Hilbert_space" title="Hilbert space">Hilbert</a></li>

<li><a href="H%C3%B6lder_space" class="mw-redirect" title="Hölder space">Hölder</a></li>

<li><a href="Nuclear_space" title="Nuclear space">Nuclear</a></li>

<li><a href="Orlicz_space" title="Orlicz space">Orlicz</a></li>

<li><a href="Schwartz_space" title="Schwartz space">Schwartz</a></li>

<li><a href="Sobolev_space" title="Sobolev space">Sobolev</a></li>

<li><a href="Topological_vector_space" title="Topological vector space">Topological vector</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Barrelled_space" title="Barrelled space">Barrelled</a></li>

<li><a href="Complete_topological_vector_space" title="Complete topological vector space">Complete</a></li>

<li><a href="Dual_space" title="Dual space">Dual</a> (<a href="Dual_space#Algebraic_dual_space" title="Dual space">Algebraic</a> / <a href="Dual_space#Continuous_dual_space" title="Dual space">Topological</a>)</li>

<li><a href="Locally_convex_topological_vector_space" title="Locally convex topological vector space">Locally convex</a></li>

<li><a href="Reflexive_space" title="Reflexive space">Reflexive</a></li>

<li><a href="Separable_space" title="Separable space">Separable</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Hahn%E2%80%93Banach_theorem" title="Hahn–Banach theorem">Hahn–Banach</a></li>

<li><a href="Riesz_representation_theorem" title="Riesz representation theorem">Riesz representation</a></li>

<li><a href="Closed_graph_theorem_(functional_analysis)" title="Closed graph theorem (functional analysis)">Closed graph</a></li>

<li><a href="Uniform_boundedness_principle" title="Uniform boundedness principle">Uniform boundedness principle</a></li>

<li><a href="Kakutani_fixed-point_theorem#Infinite-dimensional_generalizations" title="Kakutani fixed-point theorem">Kakutani fixed-point</a></li>

<li><a href="Krein%E2%80%93Milman_theorem" title="Krein–Milman theorem">Krein–Milman</a></li>

<li><a href="Gelfand%E2%80%93Naimark_theorem" title="Gelfand–Naimark theorem">Gelfand–Naimark</a></li>

<li><a href="Banach%E2%80%93Alaoglu_theorem" title="Banach–Alaoglu theorem">Banach–Alaoglu</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Operators</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Adjoint_operator" class="mw-redirect" title="Adjoint operator">Adjoint</a></li>

<li><a href="Bounded_operator" title="Bounded operator">Bounded</a></li>

<li><a href="Compact_operator" title="Compact operator">Compact</a></li>

<li><a href="Hilbert%E2%80%93Schmidt_operator" title="Hilbert–Schmidt operator">Hilbert–Schmidt</a></li>

<li><a href="Normal_operator" title="Normal operator">Normal</a></li>

<li><a href="Nuclear_operator" title="Nuclear operator">Nuclear</a></li>

<li><a href="Trace_class" title="Trace class">Trace class</a></li>

<li><a href="Transpose_of_a_linear_map" title="Transpose of a linear map">Transpose</a></li>

<li><a href="Unbounded_operator" title="Unbounded operator">Unbounded</a></li>

<li><a href="Unitary_operator" title="Unitary operator">Unitary</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Algebras</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Banach_algebra" title="Banach algebra">Banach algebra</a></li>

<li><a href="C*-algebra" title="C*-algebra">C*-algebra</a></li>

<li><a href="Spectrum_of_a_C*-algebra" title="Spectrum of a C*-algebra">Spectrum of a C*-algebra</a></li>

<li><a href="Operator_algebra" title="Operator algebra">Operator algebra</a></li>

<li><a href="Group_algebra_of_a_locally_compact_group" title="Group algebra of a locally compact group">Group algebra of a locally compact group</a></li>

<li><a href="Von_Neumann_algebra" title="Von Neumann algebra">Von Neumann algebra</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Open problems</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Invariant_subspace_problem" title="Invariant subspace problem">Invariant subspace problem</a></li>

<li><a href="Mahler's_conjecture" class="mw-redirect" title="Mahler's conjecture">Mahler's conjecture</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Hardy_space" title="Hardy space">Hardy space</a></li>

<li><a href="Spectral_theory_of_ordinary_differential_equations" title="Spectral theory of ordinary differential equations">Spectral theory of ordinary differential equations</a></li>

<li><a href="Heat_kernel" title="Heat kernel">Heat kernel</a></li>

<li><a href="Index_theorem" class="mw-redirect" title="Index theorem">Index theorem</a></li>

<li><a href="Calculus_of_variations" title="Calculus of variations">Calculus of variations</a></li>

<li><a href="Functional_calculus" title="Functional calculus">Functional calculus</a></li>

<li><a href="Integral_linear_operator" title="Integral linear operator">Integral linear operator</a></li>

<li><a href="Jones_polynomial" title="Jones polynomial">Jones polynomial</a></li>

<li><a href="Topological_quantum_field_theory" title="Topological quantum field theory">Topological quantum field theory</a></li>

<li><a href="Noncommutative_geometry" title="Noncommutative geometry">Noncommutative geometry</a></li>

<li><a href="Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a></li>

<li><a href="Distribution_(mathematics)" title="Distribution (mathematics)">Distribution</a> (or <a href="Generalized_function" title="Generalized function">Generalized functions</a>)</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Advanced topics</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Approximation_property" title="Approximation property">Approximation property</a></li>

<li><a href="Balanced_set" title="Balanced set">Balanced set</a></li>

<li><a href="Choquet_theory" title="Choquet theory">Choquet theory</a></li>

<li><a href="Weak_topology" title="Weak topology">Weak topology</a></li>

<li><a href="Banach%E2%80%93Mazur_distance" class="mw-redirect" title="Banach–Mazur distance">Banach–Mazur distance</a></li>

<li><a href="Tomita%E2%80%93Takesaki_theory" title="Tomita–Takesaki theory">Tomita–Takesaki theory</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li> Category</li></ul>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Topological_vector_spaces_(TVSs)267" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Topological_vector_spaces_(TVSs)267" style="font-size:114%;margin:0 4em"><a href="Topological_vector_space" title="Topological vector space">Topological vector spaces</a> (TVSs)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Banach_space" title="Banach space">Banach space</a></li>

<li><a href="Metrizable_topological_vector_space" title="Metrizable topological vector space">Completeness</a></li>

<li><a href="Continuous_linear_operator" title="Continuous linear operator">Continuous linear operator</a></li>

<li><a href="Linear_form" title="Linear form">Linear functional</a></li>

<li><a href="Fr%C3%A9chet_space" title="Fréchet space">Fréchet space</a></li>

<li><a href="Linear_map" title="Linear map">Linear map</a></li>

<li><a href="Locally_convex_topological_vector_space" title="Locally convex topological vector space">Locally convex space</a></li>

<li><a href="Metrizable_topological_vector_space" title="Metrizable topological vector space">Metrizability</a></li>

<li><a href="Operator_topologies" title="Operator topologies">Operator topologies</a></li>

<li><a href="Topological_vector_space" title="Topological vector space">Topological vector space</a></li>

<li><a href="Vector_space" title="Vector space">Vector space</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Main results</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Anderson%E2%80%93Kadec_theorem" title="Anderson–Kadec theorem">Anderson–Kadec</a></li>

<li><a href="Banach%E2%80%93Alaoglu_theorem" title="Banach–Alaoglu theorem">Banach–Alaoglu</a></li>

<li><a href="Closed_graph_theorem_(functional_analysis)" title="Closed graph theorem (functional analysis)">Closed graph theorem</a></li>

<li><a href="F._Riesz's_theorem" title="F. Riesz's theorem">F. Riesz's</a></li>

<li><a href="Hahn%E2%80%93Banach_theorem" title="Hahn–Banach theorem">Hahn–Banach</a> (<a href="Hyperplane_separation_theorem" title="Hyperplane separation theorem">hyperplane separation</a></li>

<li><a href="Vector-valued_Hahn%E2%80%93Banach_theorems" title="Vector-valued Hahn–Banach theorems">Vector-valued Hahn–Banach</a>)</li>

<li><a href="Open_mapping_theorem_(functional_analysis)" title="Open mapping theorem (functional analysis)">Open mapping (Banach–Schauder)</a>

<ul><li><a href="Bounded_inverse_theorem" class="mw-redirect" title="Bounded inverse theorem">Bounded inverse</a></li></ul></li>

<li><a href="Uniform_boundedness_principle" title="Uniform boundedness principle">Uniform boundedness (Banach–Steinhaus)</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Maps</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bilinear_operator" class="mw-redirect" title="Bilinear operator">Bilinear operator</a>

<li><a href="Linear_map" title="Linear map">Linear map</a>

<ul><li><a href="Almost_open_linear_map" class="mw-redirect" title="Almost open linear map">Almost open</a></li>

<li><a href="Bounded_operator" title="Bounded operator">Bounded</a></li>

<li><a href="Continuous_linear_operator" title="Continuous linear operator">Continuous</a></li>

<li><a href="Closed_linear_operator" title="Closed linear operator">Closed</a></li>

<li><a href="Compact_operator" title="Compact operator">Compact</a></li>

<li><a href="Densely_defined_operator" title="Densely defined operator">Densely defined</a></li>

<li><a href="Discontinuous_linear_map" title="Discontinuous linear map">Discontinuous</a></li></ul></li>

<li><a href="Topological_homomorphism" title="Topological homomorphism">Topological homomorphism</a></li>

<li><a href="Functional_(mathematics)" title="Functional (mathematics)">Functional</a>

<ul><li><a href="Linear_form" title="Linear form">Linear</a></li>

<li><a href="Bilinear_form" title="Bilinear form">Bilinear</a></li>

<li><a href="Sesquilinear_form" title="Sesquilinear form">Sesquilinear</a></li></ul></li>

<li><a href="Seminorm" title="Seminorm">Seminorm</a></li>

<li><a href="Transpose_of_a_linear_map" title="Transpose of a linear map">Transpose</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of sets</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Absolutely_convex_set" title="Absolutely convex set">Absolutely convex/disk</a></li>

<li><a href="Absorbing_set" title="Absorbing set">Absorbing/Radial</a></li>

<li><a href="Affine_space" title="Affine space">Affine</a></li>

<li><a href="Balanced_set" title="Balanced set">Balanced/Circled</a></li>

<li><a href="Auxiliary_normed_space" title="Auxiliary normed space">Banach disks</a></li>

<li><a href="Bounding_point" title="Bounding point">Bounding points</a></li>

<li><a href="Bounded_set_(topological_vector_space)" title="Bounded set (topological vector space)">Bounded</a></li>

<li><a href="Complemented_subspace" title="Complemented subspace">Complemented subspace</a></li>

<li><a href="Convex_set" title="Convex set">Convex</a></li>

<li><a href="Convex_cone" title="Convex cone">Convex cone (subset)</a></li>

<li><a href="Cone_(linear_algebra)" class="mw-redirect" title="Cone (linear algebra)">Linear cone (subset)</a></li>

<li><a href="Extreme_point" title="Extreme point">Extreme point</a></li>

<li><a href="Totally_bounded_space#Topological_vector_spaces" title="Totally bounded space">Pre-compact/Totally bounded</a></li>

<li><a href="Prevalent_and_shy_sets" title="Prevalent and shy sets">Prevalent/Shy</a></li>

<li><a href="Radial_set" title="Radial set">Radial</a></li>

<li><a href="Star_domain" title="Star domain">Radially convex/Star-shaped</a></li>

<li><a href="Symmetric_set" title="Symmetric set">Symmetric</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Set operations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Affine_hull" title="Affine hull">Affine hull</a></li>

<li>(<a href="Algebraic_interior#Relative_algebraic_interior" title="Algebraic interior">Relative</a>) <a href="Algebraic_interior" title="Algebraic interior">Algebraic interior (core)</a></li>

<li><a href="Convex_hull" title="Convex hull">Convex hull</a></li>

<li><a href="Linear_span" title="Linear span">Linear span</a></li>

<li><a href="Minkowski_addition" title="Minkowski addition">Minkowski addition</a></li>

<li><a href="Polar_set" title="Polar set">Polar</a></li>

<li>(<a href="Algebraic_interior#Quasi_relative_interior" title="Algebraic interior">Quasi</a>) <a href="Algebraic_interior#Relative_interior" title="Algebraic interior">Relative interior</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of TVSs</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Asplund_space" title="Asplund space">Asplund</a></li>

<li><a href="Ptak_space" title="Ptak space">B-complete/Ptak</a></li>

<li><a href="Banach_space" title="Banach space">Banach</a></li>

<li>(<a href="Countably_barrelled_space" title="Countably barrelled space">Countably</a>) <a href="Barrelled_space" title="Barrelled space">Barrelled</a></li>

<li><a href="BK-space" title="BK-space">BK-space</a></li>

<li>(<a href="Ultrabornological_space" title="Ultrabornological space">Ultra-</a>) <a href="Bornological_space" title="Bornological space">Bornological</a></li>

<li><a href="Brauner_space" title="Brauner space">Brauner</a></li>

<li><a href="Complete_topological_vector_space" title="Complete topological vector space">Complete</a></li>

<li><a href="Convenient_vector_space" title="Convenient vector space">Convenient</a></li>

<li><a href="DF-space" title="DF-space">(DF)-space</a></li>

<li><a href="Distinguished_space" title="Distinguished space">Distinguished</a></li>

<li><a href="F-space" title="F-space">F-space</a></li>

<li><a href="FK-AK_space" title="FK-AK space">FK-AK space</a></li>

<li><a href="FK-space" title="FK-space">FK-space</a></li>

<li><a href="Fr%C3%A9chet_space" title="Fréchet space">Fréchet</a>

<ul><li><a href="Differentiation_in_Fr%C3%A9chet_spaces#Tame_Fréchet_spaces" title="Differentiation in Fréchet spaces">tame Fréchet</a></li></ul></li>

<li><a href="Grothendieck_space" title="Grothendieck space">Grothendieck</a></li>

<li><a href="Hilbert_space" title="Hilbert space">Hilbert</a></li>

<li><a href="Infrabarreled_space" class="mw-redirect" title="Infrabarreled space">Infrabarreled</a></li>

<li><a href="Interpolation_space" title="Interpolation space">Interpolation space</a></li>

<li><a href="K-space_(functional_analysis)" title="K-space (functional analysis)">K-space</a></li>

<li><a href="LB-space" title="LB-space">LB-space</a></li>

<li><a href="LF-space" title="LF-space">LF-space</a></li>

<li><a href="Locally_convex_topological_vector_space" title="Locally convex topological vector space">Locally convex space</a></li>

<li><a href="Mackey_space" title="Mackey space">Mackey</a></li>

<li><a href="Metrizable_topological_vector_space" title="Metrizable topological vector space">(Pseudo)Metrizable</a></li>

<li><a href="Montel_space" title="Montel space">Montel</a></li>

<li><a href="Quasibarrelled_space" class="mw-redirect" title="Quasibarrelled space">Quasibarrelled</a></li>

<li><a href="Quasi-complete" class="mw-redirect" title="Quasi-complete">Quasi-complete</a></li>

<li><a href="Quasinorm" title="Quasinorm">Quasinormed</a></li>

<li>(<a href="Polynomially_reflexive_space" title="Polynomially reflexive space">Polynomially</a></li>

<li><a href="Semi-reflexive_space" title="Semi-reflexive space">Semi-</a>) <a href="Reflexive_space" title="Reflexive space">Reflexive</a></li>

<li><a href="Riesz_space" title="Riesz space">Riesz</a></li>

<li><a href="Schwartz_TVS" class="mw-redirect" title="Schwartz TVS">Schwartz</a></li>

<li><a href="Semi-complete" class="mw-redirect" title="Semi-complete">Semi-complete</a></li>

<li><a href="Smith_space" title="Smith space">Smith</a></li>

<li><a href="Stereotype_space" class="mw-redirect" title="Stereotype space">Stereotype</a></li>

<li><a href="Strictly_convex_space" title="Strictly convex space">Strictly</a></li>

<li><a href="Uniformly_convex_space" title="Uniformly convex space">Uniformly</a>) convex</li>

<li>(<a href="Quasi-ultrabarrelled_space" title="Quasi-ultrabarrelled space">Quasi-</a>) <a href="Ultrabarrelled_space" title="Ultrabarrelled space">Ultrabarrelled</a></li>

<li><a href="Uniformly_smooth_space" title="Uniformly smooth space">Uniformly smooth</a></li>

<li><a href="Webbed_space" title="Webbed space">Webbed</a></li>

<li><a href="Approximation_property" title="Approximation property">With the approximation property</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li> Category</li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-04-18" href="https://en.wikipedia.org/wiki/?title=Sublinear_function&oldid=1286245210">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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