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<title>Function space</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Function space</span></span>

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</style><table class="sidebar nomobile nowraplinks"><tbody><tr><th class="sidebar-title" style="letter-spacing:0.0125em; background-color:#FFCC99"><a href="Function_(mathematics)" title="Function (mathematics)">Function</a></th></tr><tr><td class="sidebar-image"><span class="texhtml texhtml-big" style="font-size:250%;"><i>x</i> ↦ <i>f</i> (<i>x</i>)</span></td></tr><tr><th class="sidebar-heading" style="font-size: 117%; letter-spacing: 0.0125em; font-weight: 500; border-top: 1px solid black; padding: 5px 0 3px">

<a href="History_of_the_function_concept" title="History of the function concept">History of the function concept</a></th></tr><tr><th class="sidebar-heading" style="font-size: 117%; letter-spacing: 0.0125em; font-weight: 500; border-top: 1px solid black; padding: 5px 0 3px">

Types by <a href="Domain_of_a_function" title="Domain of a function">domain</a> and <a href="Codomain" title="Codomain">codomain</a></th></tr><tr><td class="sidebar-content">

→ <span title="arbitrary set"><var>X</var></span></span></a></li>

→ <span title="Codomain of natural numbers"><var>X</var></span></span></a></li>

→ <span title="integers">ℤ</span></span></a></li>

→ <span title="arbitrary set"><var>X</var></span></span></a></li>

→ <span title="real numbers">ℝ</span></span></a></li>

→ <span title="arbitrary set"><var>X</var></span></span></a></li>

→ <span title="arbitrary set"><var>X</var></span></span></a></li>

→ <span title="complex numbers">ℂ</span></span></a></li>

→ <span title="arbitrary set"><var>X</var></span></span></a></li>

→ <span title="arbitrary set"><var>X</var></span></span></a></li></ul>

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<a href="List_of_types_of_functions" title="List of types of functions">Classes/properties</a> </th></tr><tr><td class="sidebar-content">

<ul><li><a href="Constant_function" title="Constant function">Constant</a></li>

<li><a href="Identity_function" title="Identity function">Identity</a></li>

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

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

<li><a href="Rational_function" title="Rational function">Rational</a></li>

<li><a href="Algebraic_function" title="Algebraic function">Algebraic</a></li>

<li><a href="Analytic_function" title="Analytic function">Analytic</a></li>

<li><a href="Smooth_function" class="mw-redirect" title="Smooth function">Smooth</a></li>

<li><a href="Continuous_function" title="Continuous function">Continuous</a></li>

<li><a href="Measurable_function" title="Measurable function">Measurable</a></li>

<li><a href="Injective_function" title="Injective function">Injective</a></li>

<li><a href="Surjective_function" title="Surjective function">Surjective</a></li>

<li><a href="Bijection" title="Bijection">Bijective</a></li></ul>

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Constructions</th></tr><tr><td class="sidebar-content">

<ul><li><a href="Restriction_(mathematics)" title="Restriction (mathematics)">Restriction</a></li>

<li><a href="Function_composition" title="Function composition">Composition</a></li>

<li><a href="Inverse_function" title="Inverse function">Inverse</a></li></ul>

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Generalizations </th></tr><tr><td class="sidebar-content">

<ul><li><a href="Relation_(mathematics)" title="Relation (mathematics)">Relation</a> (<a href="Binary_relation" title="Binary relation">Binary relation</a>)</li>

<li><a href="Set-valued_function" title="Set-valued function">Set-valued</a></li>

<li><a href="Multivalued_function" title="Multivalued function">Multivalued</a></li>

<li><a href="Partial_function" title="Partial function">Partial</a></li>

<li><a href="Implicit_function" title="Implicit function">Implicit</a></li>

<li><a href="Higher-order_function" title="Higher-order function">Higher-order</a></li>

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

<li><a href="Functor" title="Functor">Functor</a></li></ul>

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<a href="List_of_mathematical_functions" title="List of mathematical functions">List of specific functions</a></th></tr><tr><td class="sidebar-navbar"><style data-mw-deduplicate="TemplateStyles:r1239400231">

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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, a <b>function space</b> is a <a href="Set_(mathematics)" title="Set (mathematics)">set</a> of <a href="Function_(mathematics)" title="Function (mathematics)">functions</a> between two fixed sets. Often, the <a href="Domain_of_a_function" title="Domain of a function">domain</a> and/or <a href="Codomain" title="Codomain">codomain</a> will have additional <a href="Mathematical_structure" title="Mathematical structure">structure</a> which is inherited by the function space. For example, the set of functions from any set <var style="padding-right: 1px;">X</var> into a <a href="Vector_space" title="Vector space">vector space</a> has a <a href="List_of_mathematical_jargon" class="mw-redirect" title="List of mathematical jargon">natural</a> vector space structure given by <a href="Pointwise" title="Pointwise">pointwise</a> addition and scalar multiplication. In other scenarios, the function space might inherit a <a href="Topological_space" title="Topological space">topological</a> or <a href="Metric_space" title="Metric space">metric</a> structure, hence the name function <i>space</i>.

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<div class="mw-heading mw-heading2"><h2 id="In_linear_algebra">In linear algebra</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="Vector_space#Function_spaces" title="Vector space">Vector space § Function spaces</a></div>

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<p>Let <var style="padding-right: 1px;">F</var> be a <a href="Field_(mathematics)" title="Field (mathematics)">field</a> and let <var style="padding-right: 1px;">X</var> be any set. The functions <var style="padding-right: 1px;">X</var> → <var style="padding-right: 1px;">F</var> can be given the structure of a vector space over <var style="padding-right: 1px;">F</var> where the operations are defined pointwise, that is, for any <var style="padding-right: 1px;">f</var>, <var style="padding-right: 1px;">g</var> : <var style="padding-right: 1px;">X</var> → <var style="padding-right: 1px;">F</var>, any <var style="padding-right: 1px;">x</var> in <var style="padding-right: 1px;">X</var>, and any <var style="padding-right: 1px;">c</var> in <var style="padding-right: 1px;">F</var>, define

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(f+g)(x)&=f(x)+g(x)\\(c\cdot f)(x)&=c\cdot f(x)\end{aligned}}}</annotation>

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When the domain <var style="padding-right: 1px;">X</var> has additional structure, one might consider instead the <a href="Subset" title="Subset">subset</a> (or <a href="Linear_subspace" title="Linear subspace">subspace</a>) of all such functions which respect that structure. For example, if <var style="padding-right: 1px;">V</var> and also <var style="padding-right: 1px;">X</var> itself are vector spaces over <var style="padding-right: 1px;">F</var>, the set of <a href="Linear_map" title="Linear map">linear maps</a> <var style="padding-right: 1px;">X</var> → <var style="padding-right: 1px;">V</var> form a vector space over <var style="padding-right: 1px;">F</var> with pointwise operations (often denoted <a href="Hom_set" class="mw-redirect" title="Hom set">Hom</a>(<var style="padding-right: 1px;">X</var>,<var style="padding-right: 1px;">V</var>)). One such space is the <a href="Dual_space" title="Dual space">dual space</a> of <var style="padding-right: 1px;">X</var>: the set of <a href="Linear_form" title="Linear form">linear functionals</a> <var style="padding-right: 1px;">X</var> → <var style="padding-right: 1px;">F</var> with addition and scalar multiplication defined pointwise.

</p><p>The cardinal <a href="Dimension" title="Dimension">dimension</a> of a function space with no extra structure can be found by the <a href="Erd%C5%91s%E2%80%93Kaplansky_theorem" title="Erdős–Kaplansky theorem">Erdős–Kaplansky theorem</a>.

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

<p>Function spaces appear in various areas of mathematics:

</p>

<ul><li>In <a href="Set_theory" title="Set theory">set theory</a>, the set of functions from <i>X</i> to <i>Y</i> may be denoted {<i>X</i> → <i>Y</i>} or <i>Y</i><sup><i>X</i></sup>.

<ul><li>As a special case, the <a href="Power_set" title="Power set">power set</a> of a set <i>X</i> may be identified with the set of all functions from <i>X</i> to {0, 1}, denoted 2<sup><i>X</i></sup>.</li></ul></li>

<li>The set of <a href="Bijection" title="Bijection">bijections</a> from <i>X</i> to <i>Y</i> is denoted <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\leftrightarrow Y}">

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

</semantics>

</math></span><img src="./96b69b60af3edb830806b7a11982101e7facaa3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.367ex; height:2.176ex;" alt="{\displaystyle X\leftrightarrow Y}" loading="lazy"></span>. The factorial notation <i>X</i>! may be used for permutations of a single set <i>X</i>.</li>

<li>In <a href="Functional_analysis" title="Functional analysis">functional analysis</a>, the same is seen for <a href="Continuous_function" title="Continuous function">continuous</a> linear transformations, including <a href="Topological_vector_space" title="Topological vector space">topologies on the vector spaces</a> in the above, and many of the major examples are function spaces carrying a <a href="Topology" title="Topology">topology</a>; the best known examples include <a href="Hilbert_space" title="Hilbert space">Hilbert spaces</a> and <a href="Banach_space" title="Banach space">Banach spaces</a>.</li>

<li>In <a href="Functional_analysis" title="Functional analysis">functional analysis</a>, the set of all functions from the <a href="Natural_number" title="Natural number">natural numbers</a> to some set <i>X</i> is called a <i><a href="Sequence_space" title="Sequence space">sequence space</a></i>. It consists of the set of all possible <a href="Sequences" class="mw-redirect" title="Sequences">sequences</a> of elements of <i>X</i>.</li>

<li>In <a href="Topology" title="Topology">topology</a>, one may attempt to put a topology on the space of continuous functions from a <a href="Topological_space" title="Topological space">topological space</a> <i>X</i> to another one <i>Y</i>, with utility depending on the nature of the spaces. A commonly used example is the <a href="Compact-open_topology" title="Compact-open topology">compact-open topology</a>, e.g. <a href="Loop_space" title="Loop space">loop space</a>. Also available is the <a href="Product_topology" title="Product topology">product topology</a> on the space of set theoretic functions (i.e. not necessarily continuous functions) <i>Y</i><sup><i>X</i></sup>. In this context, this topology is also referred to as the <a href="Topology_of_pointwise_convergence" class="mw-redirect" title="Topology of pointwise convergence">topology of pointwise convergence</a>.</li>

<li>In <a href="Algebraic_topology" title="Algebraic topology">algebraic topology</a>, the study of <a href="Homotopy_theory" title="Homotopy theory">homotopy theory</a> is essentially that of discrete invariants of function spaces;</li>

<li>In the theory of <a href="Stochastic_process" title="Stochastic process">stochastic processes</a>, the basic technical problem is how to construct a <a href="Probability_measure" title="Probability measure">probability measure</a> on a function space of <i>paths of the process</i> (functions of time);</li>

<li>In <a href="Category_theory" title="Category theory">category theory</a>, the function space is called an <a href="Exponential_object" title="Exponential object">exponential object</a> or <a href="Exponential_object" title="Exponential object">map object</a>. It appears in one way as the representation <a href="Canonical_bifunctor" class="mw-redirect" title="Canonical bifunctor">canonical bifunctor</a>; but as (single) functor, of type <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="./b68c05513b50af3f5d38201b86c984b6061149c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.116ex; height:2.843ex;" alt="{\displaystyle [X,-]}" loading="lazy"></span>, it appears as an <a href="Adjoint_functor" class="mw-redirect" title="Adjoint functor">adjoint functor</a> to a functor of type <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\times X}">

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

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</math></span><img src="./9217169749da5e8d527876e4765da9723ba7014c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.629ex; height:2.343ex;" alt="{\displaystyle -\times X}" loading="lazy"></span> on objects;</li>

<li>In <a href="Functional_programming" title="Functional programming">functional programming</a> and <a href="Lambda_calculus" title="Lambda calculus">lambda calculus</a>, <a href="Function_type" title="Function type">function types</a> are used to express the idea of <a href="Higher-order_function" title="Higher-order function">higher-order functions</a></li>

<li>In programming more generally, many <a href="Higher-order_function" title="Higher-order function">higher-order function</a> concepts occur with or without explicit typing, such as <a href="Closure_(computer_programming)" title="Closure (computer programming)">closures</a>.</li>

<li>In <a href="Domain_theory" title="Domain theory">domain theory</a>, the basic idea is to find constructions from <a href="Partial_order" class="mw-redirect" title="Partial order">partial orders</a> that can model lambda calculus, by creating a well-behaved <a href="Cartesian_closed_category" title="Cartesian closed category">Cartesian closed category</a>.</li>

<li>In the <a href="Representation_theory_of_finite_groups" title="Representation theory of finite groups">representation theory of finite groups</a>, given two finite-dimensional representations <var style="padding-right: 1px;">V</var> and <var style="padding-right: 1px;">W</var> of a group <var style="padding-right: 1px;">G</var>, one can form a representation of <var style="padding-right: 1px;">G</var> over the vector space of linear maps Hom(<var style="padding-right: 1px;">V</var>,<var style="padding-right: 1px;">W</var>) called the <a href="Hom_representation" class="mw-redirect" title="Hom representation">Hom representation</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Functional_analysis">Functional analysis</h2></div>

<p><a href="Functional_analysis" title="Functional analysis">Functional analysis</a> is organized around adequate techniques to bring function spaces as <a href="Topological_vector_space" title="Topological vector space">topological vector spaces</a> within reach of the ideas that would apply to <a href="Normed_space" class="mw-redirect" title="Normed space">normed spaces</a> of finite dimension. Here we use the real line as an example domain, but the spaces below exist on suitable open subsets <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega \subseteq \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./c2776d402b8e3ef34e3505d645b264f6dfb6a72e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.673ex; height:2.509ex;" alt="{\displaystyle \Omega \subseteq \mathbb {R} ^{n}}" loading="lazy"></span>

</p>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./b7c83d1af403f8fe4ad3b88bf35451369497d8c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.254ex; height:2.843ex;" alt="{\displaystyle C(\mathbb {R} )}" loading="lazy"></span> <a href="Continuous_functions" class="mw-redirect" title="Continuous functions">continuous functions</a> endowed with the <a href="Uniform_norm" title="Uniform norm">uniform norm</a> topology</li>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./c9473f76ade23775cc0ddfee38381c14acc71c75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.093ex; height:2.843ex;" alt="{\displaystyle C_{c}(\mathbb {R} )}" loading="lazy"></span> continuous functions with <a href="Support_(mathematics)#Compact_support" title="Support (mathematics)">compact support</a></li>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./df9abf2d8d1b9e447e49ed4cc833af5d56d698d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.251ex; height:2.843ex;" alt="{\displaystyle B(\mathbb {R} )}" loading="lazy"></span> <a href="Bounded_function" title="Bounded function">bounded functions</a></li>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./b291a758a02ce8473dba27a2bcc64085902da8bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.203ex; height:2.843ex;" alt="{\displaystyle C_{0}(\mathbb {R} )}" loading="lazy"></span> continuous functions which vanish at infinity</li>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C^{r}(\mathbb {R} )}</annotation>

</semantics>

</math></span><img src="./be46f348c7b663b612bf153fd1825bd51cbc2632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.259ex; height:2.843ex;" alt="{\displaystyle C^{r}(\mathbb {R} )}" loading="lazy"></span> continuous functions that have <i>r</i> continuous derivatives.</li>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C^{\infty }(\mathbb {R} )}</annotation>

</semantics>

</math></span><img src="./c913ced284ad05bd42c8f004a3a83dee6323cb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.161ex; height:2.843ex;" alt="{\displaystyle C^{\infty }(\mathbb {R} )}" loading="lazy"></span> <a href="Smooth_functions" class="mw-redirect" title="Smooth functions">smooth functions</a></li>

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{c}^{\infty }(\mathbb {R} )}</annotation>

</semantics>

</math></span><img src="./dbbf3b0bd3006d257867fee68628fcdd300e2f04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.161ex; height:2.843ex;" alt="{\displaystyle C_{c}^{\infty }(\mathbb {R} )}" loading="lazy"></span> <a href="Smooth_functions" class="mw-redirect" title="Smooth functions">smooth functions</a> with <a href="Support_(mathematics)#Compact_support" title="Support (mathematics)">compact support</a> (i.e. the set of <a href="Bump_function" title="Bump function">bump functions</a>)</li>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C^{\omega }(\mathbb {R} )}</annotation>

</semantics>

</math></span><img src="./41b09f7854c08a7e4103a4e2fdd65414e3c6d06d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.54ex; height:2.843ex;" alt="{\displaystyle C^{\omega }(\mathbb {R} )}" loading="lazy"></span> <a href="Analytic_function" title="Analytic function">real analytic functions</a></li>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./81b25cc9016efea65c3a2be0b1a358b0d399ce3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.129ex; height:2.843ex;" alt="{\displaystyle L^{p}(\mathbb {R} )}" loading="lazy"></span>, for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq p\leq \infty }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1\leq p\leq \infty }</annotation>

</semantics>

</math></span><img src="./b7d63bb8c8def80f4fb709fbb2aae6a11c9cd41d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.853ex; height:2.509ex;" alt="{\displaystyle 1\leq p\leq \infty }" loading="lazy"></span>, is the <a href="Lp_space" title="Lp space">L<sup>p</sup> space</a> of <a href="Measurable_function" title="Measurable function">measurable</a> functions whose <i>p</i>-norm <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \|f\|_{p}=\left(\int _{\mathbb {R} }|f|^{p}\right)^{1/p}}">

<msub>

</mrow>

</msub>

<msup>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \|f\|_{p}=\left(\int _{\mathbb {R} }|f|^{p}\right)^{1/p}}</annotation>

</semantics>

</math></span><img src="./9e60dfd0bdc0e6b992d0eeef1b184467f9b54cb5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.129ex; height:3.843ex;" alt="{\textstyle \|f\|_{p}=\left(\int _{\mathbb {R} }|f|^{p}\right)^{1/p}}" loading="lazy"></span> is finite</li>

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./0751119ebd490eacaceee66c787196ffae55c316.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.98ex; height:2.843ex;" alt="{\displaystyle {\mathcal {S}}(\mathbb {R} )}" loading="lazy"></span>, the <a href="Schwartz_space" title="Schwartz space">Schwartz space</a> of <a href="Rapidly_decreasing" class="mw-redirect" title="Rapidly decreasing">rapidly decreasing</a> <a href="Smooth_functions" class="mw-redirect" title="Smooth functions">smooth functions</a> and its continuous dual, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}'(\mathbb {R} )}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}'(\mathbb {R} )}</annotation>

</semantics>

</math></span><img src="./ec8c5377581a431b919b6170e94f771d69583a88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.69ex; height:3.009ex;" alt="{\displaystyle {\mathcal {S}}'(\mathbb {R} )}" loading="lazy"></span> <a href="Tempered_distributions" class="mw-redirect" title="Tempered distributions">tempered distributions</a></li>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./e87480d9aeee93a7a4f68d95da878998d402bc4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.412ex; height:2.843ex;" alt="{\displaystyle D(\mathbb {R} )}" loading="lazy"></span> compact support in limit topology</li>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./b59dd87b72e0af403c6b5f6e53b71ca14a83d4d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.881ex; height:2.676ex;" alt="{\displaystyle W^{k,p}}" loading="lazy"></span> <a href="Sobolev_space" title="Sobolev space">Sobolev space</a> of functions whose <a href="Weak_derivative" title="Weak derivative">weak derivatives</a> up to order <i>k</i> are in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./cf2317aaca1ecee4b8ccf667bc1001059eae5850.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.642ex; height:2.343ex;" alt="{\displaystyle L^{p}}" loading="lazy"></span></li>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./c9a9cef2217740befda1b31bb9a65ec7cb8a1d8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.343ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{U}}" loading="lazy"></span> holomorphic functions</li>

<li>linear functions</li>

<li>piecewise linear functions</li>

<li>continuous functions, compact open topology</li>

<li>all functions, space of pointwise convergence</li>

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

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

<li><a href="C%C3%A0dl%C3%A0g" title="Càdlàg">Càdlàg</a> functions, also known as the <a href="Anatoliy_Skorokhod" title="Anatoliy Skorokhod">Skorokhod</a> space</li>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{Lip}}_{0}(\mathbb {R} )}</annotation>

</semantics>

</math></span><img src="./04a055d6bcd80b0a650d23dd5d47ae6af68a07e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.934ex; height:2.843ex;" alt="{\displaystyle {\text{Lip}}_{0}(\mathbb {R} )}" loading="lazy"></span>, the space of all <a href="Lipschitz_continuous" class="mw-redirect" title="Lipschitz continuous">Lipschitz</a> functions on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> that vanish at zero.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Uniform_norm">Uniform norm</h2></div>

<p>If <span class="texhtml"><i>y</i></span> is an element of the function space <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}(a,b)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}(a,b)}</annotation>

</semantics>

</math></span><img src="./d89418b6b41c85096bfadf1f877d9113642270f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.31ex; height:2.843ex;" alt="{\displaystyle {\mathcal {C}}(a,b)}" loading="lazy"></span> of all <a href="Continuous_function" title="Continuous function">continuous functions</a> that are defined on a <a href="Closed_interval" class="mw-redirect" title="Closed interval">closed interval</a> <span class="texhtml">[<i>a</i>, <i>b</i>]</span>, the <b><a href="Norm_(mathematics)" title="Norm (mathematics)">norm</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|y\|_{\infty }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./7b93729fac75d797d1c804f37d0535f7d59e293f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.356ex; height:2.843ex;" alt="{\displaystyle \|y\|_{\infty }}" loading="lazy"></span></b> defined on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}(a,b)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}(a,b)}</annotation>

</semantics>

</math></span><img src="./d89418b6b41c85096bfadf1f877d9113642270f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.31ex; height:2.843ex;" alt="{\displaystyle {\mathcal {C}}(a,b)}" loading="lazy"></span> is the maximum <a href="Absolute_value" title="Absolute value">absolute value</a> of <span class="texhtml"><i>y</i> (<i>x</i>)</span> for <span class="texhtml"><i>a</i> ≤ <i>x</i> ≤ <i>b</i></span>,<sup id="cite_ref-GelfandFominP6_2-0" class="reference"><a href="#cite_note-GelfandFominP6-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

<msub>

</mrow>

</msub>

</mrow>

</munder>

</mrow>

</mrow>

<mtext>where</mtext>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|y\|_{\infty }\equiv \max _{a\leq x\leq b}|y(x)|\qquad {\text{where}}\ \ y\in {\mathcal {C}}(a,b)}</annotation>

</semantics>

</math></span></span>

</p><p>is called the <i><a href="Uniform_norm" title="Uniform norm">uniform norm</a></i> or <i>supremum norm</i> ('sup norm').

</p>

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

<ul><li>Kolmogorov, A. N., & Fomin, S. V. (1967). Elements of the theory of functions and functional analysis. Courier Dover Publications.</li>

<li>Stein, Elias; Shakarchi, R. (2011). Functional Analysis: An Introduction to Further Topics in Analysis. Princeton University Press.</li></ul>

<ul><li><a href="List_of_mathematical_functions" title="List of mathematical functions">List of mathematical functions</a></li>

<li><a href="Clifford_algebra" title="Clifford algebra">Clifford algebra</a></li>

<li><a href="Tensor_field" title="Tensor field">Tensor field</a></li>

<li><a href="Spectral_theory" title="Spectral theory">Spectral theory</a></li>

<li><a href="Functional_determinant" title="Functional determinant">Functional determinant</a></li></ul>

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

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</style><cite id="CITEREFFultonHarris1991" class="citation book cs1">Fulton, William; Harris, Joe (1991). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=6GUH8ARxhp8C"><i>Representation Theory: A First Course</i></a>. Springer Science & Business Media. p. 4. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780387974958</bdi>.</cite></span>

</li>

<li id="cite_note-GelfandFominP6-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-GelfandFominP6_2-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFGelfandFomin2000" class="citation book cs1"><a href="Israel_Gelfand" title="Israel Gelfand">Gelfand, I. M.</a>; <a href="Sergei_Fomin" title="Sergei Fomin">Fomin, S. V.</a> (2000). Silverman, Richard A. (ed.). <a rel="nofollow" class="external text" href="http://store.doverpublications.com/0486414485.html"><i>Calculus of variations</i></a> (Unabridged repr. ed.). Mineola, New York: Dover Publications. p. 6. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0486414485</bdi>.</cite></span>

</li>

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</style></div><div role="navigation" class="navbox authority-control" aria-label="Navbox780" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Authority control databases: National </th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://id.ndl.go.jp/auth/ndlna/00564963">Japan</a></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="prostory funkcí"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph124659&CON_LNG=ENG">Czech Republic</a></span></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://www.nli.org.il/en/authorities/987007553159205171">Israel</a></span></li></ul></div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Lp_spaces64" 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="Lp_spaces64" style="font-size:114%;margin:0 4em"><a href="Lp_space" title="Lp space">Lp spaces</a></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</a> & <a href="Hilbert_space" title="Hilbert space">Hilbert spaces</a></li>

<li><a href="Lp_space" title="Lp space"><i>L</i><sup><i>p</i></sup> spaces</a></li>

<li><a href="Measure_(mathematics)" title="Measure (mathematics)">Measure</a>

<ul><li><a href="Lebesgue_measure" title="Lebesgue measure">Lebesgue</a></li></ul></li>

<li><a href="Measure_space" title="Measure space">Measure space</a></li>

<li><a href="Measurable_space" title="Measurable space">Measurable space</a>/<a href="Measurable_function" title="Measurable function">function</a></li>

<li><a href="Minkowski_distance" title="Minkowski distance">Minkowski distance</a></li>

<li><a href="Sequence_space" title="Sequence space">Sequence spaces</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="L1_space" class="mw-redirect" title="L1 space"><i>L</i><sup>1</sup> spaces</a></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="Integrable_function" class="mw-redirect" title="Integrable function">Integrable function</a></li>

<li><a href="Lebesgue_integration" class="mw-redirect" title="Lebesgue integration">Lebesgue integration</a></li>

<li><a href="Taxicab_geometry" title="Taxicab geometry">Taxicab geometry</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="L2_space" class="mw-redirect" title="L2 space"><i>L</i><sup>2</sup> spaces</a></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="Bessel's_inequality" title="Bessel's inequality">Bessel's</a></li>

<li><a href="Cauchy%E2%80%93Schwarz_inequality" title="Cauchy–Schwarz inequality">Cauchy–Schwarz</a></li>

<li><a href="Euclidean_distance" title="Euclidean distance">Euclidean distance</a></li>

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

<li><a href="Parseval's_identity" title="Parseval's identity">Parseval's identity</a></li>

<li><a href="Polarization_identity" title="Polarization identity">Polarization identity</a></li>

<li><a href="Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a></li>

<li><a href="Square-integrable_function" title="Square-integrable function">Square-integrable function</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="L-infinity" title="L-infinity"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\infty }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./b9ab400cc4dfd865180cd84c72dc894ca457671f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.458ex; height:2.343ex;" alt="{\displaystyle L^{\infty }}" loading="lazy"></span> spaces</a></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="Bounded_function" title="Bounded function">Bounded function</a></li>

<li><a href="Chebyshev_distance" title="Chebyshev distance">Chebyshev distance</a></li>

<li><a href="Infimum_and_supremum" title="Infimum and supremum">Infimum and supremum</a>

<ul><li><a href="Essential_infimum_and_essential_supremum" title="Essential infimum and essential supremum">Essential</a></li></ul></li>

<li><a href="Uniform_norm" title="Uniform norm">Uniform norm</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="Almost_everywhere" title="Almost everywhere">Almost everywhere</a></li>

<li><a href="Convergence_almost_everywhere" class="mw-redirect" title="Convergence almost everywhere">Convergence almost everywhere</a></li>

<li><a href="Convergence_in_measure" title="Convergence in measure">Convergence in measure</a></li>

<li><a href="Integral_transform" title="Integral transform">Integral transform</a></li>

<li><a href="Locally_integrable_function" title="Locally integrable function">Locally integrable function</a></li>

<li><a href="Measurable_function" title="Measurable function">Measurable function</a></li>

<li><a href="Symmetric_decreasing_rearrangement" title="Symmetric decreasing rearrangement">Symmetric decreasing rearrangement</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Inequalities</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="Babenko%E2%80%93Beckner_inequality" title="Babenko–Beckner inequality">Babenko–Beckner</a></li>

<li><a href="Chebyshev's_inequality" title="Chebyshev's inequality">Chebyshev's</a></li>

<li><a href="Clarkson's_inequalities" title="Clarkson's inequalities">Clarkson's</a></li>

<li><a href="Hanner's_inequalities" title="Hanner's inequalities">Hanner's</a></li>

<li><a href="Hausdorff%E2%80%93Young_inequality" title="Hausdorff–Young inequality">Hausdorff–Young</a></li>

<li><a href="H%C3%B6lder's_inequality" title="Hölder's inequality">Hölder's</a></li>

<li><a href="Markov's_inequality" title="Markov's inequality">Markov's</a></li>

<li><a href="Minkowski_inequality" title="Minkowski inequality">Minkowski</a></li>

<li><a href="Young's_convolution_inequality" title="Young's convolution inequality">Young's convolution</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Results</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="Marcinkiewicz_interpolation_theorem" title="Marcinkiewicz interpolation theorem">Marcinkiewicz interpolation theorem</a></li>

<li><a href="Plancherel_theorem" title="Plancherel theorem">Plancherel theorem</a></li>

<li><a href="Riemann%E2%80%93Lebesgue_lemma" title="Riemann–Lebesgue lemma">Riemann–Lebesgue</a></li>

<li><a href="Riesz%E2%80%93Fischer_theorem" title="Riesz–Fischer theorem">Riesz–Fischer theorem</a></li>

<li><a href="Riesz%E2%80%93Thorin_theorem" title="Riesz–Thorin theorem">Riesz–Thorin theorem</a></li></ul>

</div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><span style="font-size: 85%;">For <a href="Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a></span></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="Isoperimetric_inequality" title="Isoperimetric inequality">Isoperimetric inequality</a></li>

<li><a href="Brunn%E2%80%93Minkowski_theorem" title="Brunn–Minkowski theorem">Brunn–Minkowski theorem</a>

<ul><li><a href="Milman's_reverse_Brunn%E2%80%93Minkowski_inequality" title="Milman's reverse Brunn–Minkowski inequality">Milman's reverse</a></li></ul></li>

<li><a href="Minkowski%E2%80%93Steiner_formula" title="Minkowski–Steiner formula">Minkowski–Steiner formula</a></li>

<li><a href="Pr%C3%A9kopa%E2%80%93Leindler_inequality" title="Prékopa–Leindler inequality">Prékopa–Leindler inequality</a></li>

<li><a href="Vitale's_random_Brunn%E2%80%93Minkowski_inequality" title="Vitale's random Brunn–Minkowski inequality">Vitale's random Brunn–Minkowski inequality</a></li></ul>

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

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications & related</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="Bochner_space" title="Bochner space">Bochner space</a></li>

<li><a href="Fourier_analysis" title="Fourier analysis">Fourier analysis</a></li>

<li><a href="Lorentz_space" title="Lorentz space">Lorentz space</a></li>

<li><a href="Probability_theory" title="Probability theory">Probability theory</a></li>

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

<li><a href="Real_analysis" title="Real analysis">Real analysis</a></li>

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

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

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

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

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

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Measure_theory138" 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="Measure_theory138" style="font-size:114%;margin:0 4em"><a href="Measure_theory" class="mw-redirect" title="Measure theory">Measure theory</a></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="Absolute_continuity" title="Absolute continuity">Absolute continuity</a> <a href="Absolute_continuity_(measure_theory)" class="mw-redirect" title="Absolute continuity (measure theory)">of measures</a></li>

<li><a href="Lebesgue_integration" class="mw-redirect" title="Lebesgue integration">Lebesgue integration</a></li>

<li><a href="Lp_space" title="Lp space"><i>L</i><sup><i>p</i></sup> spaces</a></li>

<li><a href="Measure_(mathematics)" title="Measure (mathematics)">Measure</a></li>

<li><a href="Measure_space" title="Measure space">Measure space</a>

<ul><li><a href="Probability_space" title="Probability space">Probability space</a></li></ul></li>

<li><a href="Measurable_space" title="Measurable space">Measurable space</a>/<a href="Measurable_function" title="Measurable function">function</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">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="Almost_everywhere" title="Almost everywhere">Almost everywhere</a></li>

<li><a href="Baire_set" title="Baire set">Baire set</a></li>

<li><a href="Borel_set" title="Borel set">Borel set</a>

<ul><li><a href="Borel_equivalence_relation" title="Borel equivalence relation">equivalence relation</a></li></ul></li>

<li><a href="Standard_Borel_space" title="Standard Borel space">Borel space</a></li>

<li><a href="Carath%C3%A9odory's_criterion" title="Carathéodory's criterion">Carathéodory's criterion</a></li>

<li><a href="Cylindrical_%CF%83-algebra" title="Cylindrical σ-algebra">Cylindrical σ-algebra</a>

<ul><li><a href="Cylinder_set" title="Cylinder set">Cylinder set</a></li></ul></li>

<li><a href="Dynkin_system" title="Dynkin system">𝜆-system</a></li>

<li><a href="Essential_range" title="Essential range">Essential range</a>

<ul><li><a href="Essential_infimum_and_essential_supremum" title="Essential infimum and essential supremum">infimum/supremum</a></li></ul></li>

<li><a href="Locally_measurable_set" class="mw-redirect" title="Locally measurable set">Locally measurable</a></li>

<li><a href="Pi-system" title="Pi-system"><span class="texhtml mvar" style="font-style:italic;">π</span>-system</a></li>

<li><a href="%CE%A3-algebra" title="Σ-algebra">σ-algebra</a></li>

<li><a href="Non-measurable_set" title="Non-measurable set">Non-measurable set</a>

<ul><li><a href="Vitali_set" title="Vitali set">Vitali set</a></li></ul></li>

<li><a href="Support_(measure_theory)" title="Support (measure theory)">Support</a></li>

<li><a href="Transverse_measure" title="Transverse measure">Transverse measure</a></li>

<li><a href="Universally_measurable_set" title="Universally measurable set">Universally measurable</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of <a href="Measure_(mathematics)" title="Measure (mathematics)">measures</a></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="Atomic_measure" class="mw-redirect" title="Atomic measure">Atomic</a></li>

<li><a href="Baire_measure" title="Baire measure">Baire</a></li>

<li><a href="Banach_measure" title="Banach measure">Banach</a></li>

<li><a href="Besov_measure" title="Besov measure">Besov</a></li>

<li><a href="Borel_measure" title="Borel measure">Borel</a></li>

<li><a href="Brown_measure" title="Brown measure">Brown</a></li>

<li><a href="Complex_measure" title="Complex measure">Complex</a></li>

<li><a href="Complete_measure" title="Complete measure">Complete</a></li>

<li><a href="Content_(measure_theory)" title="Content (measure theory)">Content</a></li>

<li>(<a href="Logarithmically_concave_measure" title="Logarithmically concave measure">Logarithmically</a>) <a href="Convex_measure" title="Convex measure">Convex</a></li>

<li><a href="Decomposable_measure" title="Decomposable measure">Decomposable</a></li>

<li><a href="Discrete_measure" title="Discrete measure">Discrete</a></li>

<li><a href="Equivalence_(measure_theory)" title="Equivalence (measure theory)">Equivalent</a></li>

<li><a href="Finite_measure" title="Finite measure">Finite</a></li>

<li><a href="Inner_measure" title="Inner measure">Inner</a></li>

<li>(<a href="Quasi-invariant_measure" title="Quasi-invariant measure">Quasi-</a>) <a href="Invariant_measure" title="Invariant measure">Invariant</a></li>

<li><a href="Locally_finite_measure" title="Locally finite measure">Locally finite</a></li>

<li><a href="Maximising_measure" title="Maximising measure">Maximising</a></li>

<li><a href="Metric_outer_measure" title="Metric outer measure">Metric outer</a></li>

<li><a href="Outer_measure" title="Outer measure">Outer</a></li>

<li><a href="Perfect_measure" title="Perfect measure">Perfect</a></li>

<li><a href="Pre-measure" title="Pre-measure">Pre-measure</a></li>

<li>(<a href="Sub-probability_measure" title="Sub-probability measure">Sub-</a>) <a href="Probability_measure" title="Probability measure">Probability</a></li>

<li><a href="Projection-valued_measure" title="Projection-valued measure">Projection-valued</a></li>

<li><a href="Radon_measure" title="Radon measure">Radon</a></li>

<li><a href="Random_measure" title="Random measure">Random</a></li>

<li><a href="Regular_measure" title="Regular measure">Regular</a>

<ul><li><a href="Borel_regular_measure" title="Borel regular measure">Borel regular</a></li>

<li><a href="Inner_regular_measure" class="mw-redirect" title="Inner regular measure">Inner regular</a></li>

<li><a href="Outer_regular_measure" class="mw-redirect" title="Outer regular measure">Outer regular</a></li></ul></li>

<li><a href="Saturated_measure" title="Saturated measure">Saturated</a></li>

<li><a href="Set_function" title="Set function">Set function</a></li>

<li><a href="%CE%A3-finite_measure" title="Σ-finite measure">σ-finite</a></li>

<li><a href="S-finite_measure" title="S-finite measure">s-finite</a></li>

<li><a href="Signed_measure" title="Signed measure">Signed</a></li>

<li><a href="Singular_measure" title="Singular measure">Singular</a></li>

<li><a href="Spectral_measure" class="mw-redirect" title="Spectral measure">Spectral</a></li>

<li><a href="Strictly_positive_measure" title="Strictly positive measure">Strictly positive</a></li>

<li><a href="Tightness_of_measures" title="Tightness of measures">Tight</a></li>

<li><a href="Vector_measure" title="Vector measure">Vector</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Particular measures</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="Counting_measure" title="Counting measure">Counting</a></li>

<li><a href="Dirac_measure" title="Dirac measure">Dirac</a></li>

<li><a href="Euler_measure" title="Euler measure">Euler</a></li>

<li><a href="Gaussian_measure" title="Gaussian measure">Gaussian</a></li>

<li><a href="Harmonic_measure" title="Harmonic measure">Harmonic</a></li>

<li><a href="Hausdorff_measure" title="Hausdorff measure">Hausdorff</a></li>

<li><a href="Intensity_measure" title="Intensity measure">Intensity</a></li>

<li><a href="Lebesgue_measure" title="Lebesgue measure">Lebesgue</a>

<ul><li><a href="Infinite-dimensional_Lebesgue_measure" title="Infinite-dimensional Lebesgue measure">Infinite-dimensional</a></li></ul></li>

<li><a href="Positive_real_numbers#Logarithmic_measure" title="Positive real numbers">Logarithmic</a></li>

<li><a href="Product_measure" title="Product measure">Product</a>

<ul><li><a href="Projection_(measure_theory)" title="Projection (measure theory)">Projections</a></li></ul></li>

<li><a href="Pushforward_measure" title="Pushforward measure">Pushforward</a></li>

<li><a href="Spherical_measure" title="Spherical measure">Spherical measure</a></li>

<li><a href="Tangent_measure" title="Tangent measure">Tangent</a></li>

<li><a href="Trivial_measure" title="Trivial measure">Trivial</a></li>

<li><a href="Young_measure" title="Young measure">Young</a></li></ul>

<ul><li><a href="Measurable_function" title="Measurable function">Measurable function</a>

<ul><li><a href="Bochner_measurable_function" title="Bochner measurable function">Bochner</a></li>

<li><a href="Strongly_measurable_function" title="Strongly measurable function">Strongly</a></li>

<li><a href="Weakly_measurable_function" title="Weakly measurable function">Weakly</a></li></ul></li>

<li>Convergence: <a href="Convergence_almost_everywhere" class="mw-redirect" title="Convergence almost everywhere">almost everywhere</a></li>

<li><a href="Convergence_of_measures" title="Convergence of measures">of measures</a></li>

<li><a href="Convergence_in_measure" title="Convergence in measure">in measure</a></li>

<li><a href="Convergence_of_random_variables" title="Convergence of random variables">of random variables</a>

<ul><li><a href="Convergence_in_distribution" class="mw-redirect" title="Convergence in distribution">in distribution</a></li>

<li><a href="Convergence_in_probability" class="mw-redirect" title="Convergence in probability">in probability</a></li></ul></li>

<li><a href="Cylinder_set_measure" title="Cylinder set measure">Cylinder set measure</a></li>

<li>Random: <a href="Random_compact_set" title="Random compact set">compact set</a></li>

<li><a href="Random_element" title="Random element">element</a></li>

<li><a href="Random_measure" title="Random measure">measure</a></li>

<li><a href="Stochastic_process" title="Stochastic process">process</a></li>

<li><a href="Random_variable" title="Random variable">variable</a></li>

<li><a href="Multivariate_random_variable" title="Multivariate random variable">vector</a></li>

<li><a href="Projection-valued_measure" title="Projection-valued measure">Projection-valued measure</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="Carath%C3%A9odory's_extension_theorem" title="Carathéodory's extension theorem">Carathéodory's extension theorem</a></li>

<li>Convergence theorems

<ul><li><a href="Dominated_convergence_theorem" title="Dominated convergence theorem">Dominated</a></li>

<li><a href="Monotone_convergence_theorem" title="Monotone convergence theorem">Monotone</a></li>

<li><a href="Vitali_convergence_theorem" title="Vitali convergence theorem">Vitali</a></li></ul></li>

<li>Decomposition theorems

<li><a href="Jordan_decomposition_theorem" class="mw-redirect" title="Jordan decomposition theorem">Jordan</a></li>

<li><a href="Maharam's_theorem" title="Maharam's theorem">Maharam's</a></li></ul></li>

<li><a href="Egorov's_theorem" title="Egorov's theorem">Egorov's</a></li>

<li><a href="Fatou's_lemma" title="Fatou's lemma">Fatou's lemma</a></li>

<li><a href="Fubini's_theorem" title="Fubini's theorem">Fubini's</a>

<ul><li><a href="Fubini%E2%80%93Tonelli_theorem" class="mw-redirect" title="Fubini–Tonelli theorem">Fubini–Tonelli</a></li></ul></li>

<li><a href="H%C3%B6lder's_inequality" title="Hölder's inequality">Hölder's inequality</a></li>

<li><a href="Minkowski_inequality" title="Minkowski inequality">Minkowski inequality</a></li>

<li><a href="Radon%E2%80%93Nikodym_theorem" title="Radon–Nikodym theorem">Radon–Nikodym</a></li>

<li><a href="Riesz%E2%80%93Markov%E2%80%93Kakutani_representation_theorem" title="Riesz–Markov–Kakutani representation theorem">Riesz–Markov–Kakutani representation theorem</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other results</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="Disintegration_theorem" title="Disintegration theorem">Disintegration theorem</a>

<ul><li><a href="Lifting_theory" title="Lifting theory">Lifting theory</a></li></ul></li>

<li><a href="Lebesgue's_density_theorem" title="Lebesgue's density theorem">Lebesgue's density theorem</a></li>

<li><a href="Lebesgue_differentiation_theorem" title="Lebesgue differentiation theorem">Lebesgue differentiation theorem</a></li>

<li><a href="Sard's_theorem" title="Sard's theorem">Sard's theorem</a></li>

<li><a href="Vitali%E2%80%93Hahn%E2%80%93Saks_theorem" title="Vitali–Hahn–Saks theorem">Vitali–Hahn–Saks theorem</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><span style="font-size: 85%;">For <a href="Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a></span></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="Isoperimetric_inequality" title="Isoperimetric inequality">Isoperimetric inequality</a></li>

<li><a href="Brunn%E2%80%93Minkowski_theorem" title="Brunn–Minkowski theorem">Brunn–Minkowski theorem</a>

<ul><li><a href="Milman's_reverse_Brunn%E2%80%93Minkowski_inequality" title="Milman's reverse Brunn–Minkowski inequality">Milman's reverse</a></li></ul></li>

<li><a href="Minkowski%E2%80%93Steiner_formula" title="Minkowski–Steiner formula">Minkowski–Steiner formula</a></li>

<li><a href="Pr%C3%A9kopa%E2%80%93Leindler_inequality" title="Prékopa–Leindler inequality">Prékopa–Leindler inequality</a></li>

<li><a href="Vitale's_random_Brunn%E2%80%93Minkowski_inequality" title="Vitale's random Brunn–Minkowski inequality">Vitale's random Brunn–Minkowski inequality</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications & related</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="Convex_analysis" title="Convex analysis">Convex analysis</a></li>

<li><a href="Descriptive_set_theory" title="Descriptive set theory">Descriptive set theory</a></li>

<li><a href="Probability_theory" title="Probability theory">Probability theory</a></li>

<li><a href="Real_analysis" title="Real analysis">Real analysis</a></li>

<li><a href="Spectral_theory" title="Spectral theory">Spectral theory</a></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 2025-06-22" href="https://en.wikipedia.org/wiki/?title=Function_space&oldid=1296804171">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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