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<title>Orthogonal complement</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Orthogonal complement</span></span>

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<p>In the <a href="Mathematics" title="Mathematics">mathematical</a> fields of <a href="Linear_algebra" title="Linear algebra">linear algebra</a> and <a href="Functional_analysis" title="Functional analysis">functional analysis</a>, the <b>orthogonal complement</b> of a <a href="Linear_subspace" title="Linear subspace">subspace</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 W}">

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

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</math></span><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"></span> of a <a href="Vector_space" title="Vector space">vector space</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 V}">

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

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</math></span><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"></span> equipped with a <a href="Bilinear_form" title="Bilinear form">bilinear form</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 B}">

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

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</math></span><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"></span> is the set <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 W^{\perp }}">

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

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</math></span><img src="./dda1dcc2d841bdc9aefcf9c90799310a56fcad33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.019ex; height:2.676ex;" alt="{\displaystyle W^{\perp }}" loading="lazy"></span> of all vectors 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 V}">

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

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</math></span><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"></span> that are <a href="Orthogonal" class="mw-redirect" title="Orthogonal">orthogonal</a> to every vector 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 W}">

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

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</math></span><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"></span>. Informally, it is called the <b>perp</b>, short for <b>perpendicular complement</b>. It is a subspace of <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 V}">

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

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</math></span><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"></span>.

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

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<annotation encoding="application/x-tex">{\displaystyle V=(\mathbb {R} ^{5},\langle \cdot ,\cdot \rangle )}</annotation>

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</math></span><img src="./23454f4c0c8db268206a44832de174af152acdca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.598ex; height:3.176ex;" alt="{\displaystyle V=(\mathbb {R} ^{5},\langle \cdot ,\cdot \rangle )}" loading="lazy"></span> be the vector space equipped with the usual <a href="Dot_product" title="Dot product">dot product</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 \langle \cdot ,\cdot \rangle }">

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

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</math></span><img src="./9a50080b735975d8001c9552ac2134b49ad534c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.137ex; height:2.843ex;" alt="{\displaystyle \langle \cdot ,\cdot \rangle }" loading="lazy"></span> (thus making it an <a href="Inner_product_space" title="Inner product space">inner product space</a>), and let <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\{\mathbf {u} \in V:\mathbf {A} x=\mathbf {u} ,\ x\in \mathbb {R} ^{2}\},}">

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<annotation encoding="application/x-tex">{\displaystyle W=\{\mathbf {u} \in V:\mathbf {A} x=\mathbf {u} ,\ x\in \mathbb {R} ^{2}\},}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} ={\begin{pmatrix}1&0\\0&1\\2&6\\3&9\\5&3\\\end{pmatrix}}.}</annotation>

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then its orthogonal complement <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W^{\perp }=\{\mathbf {v} \in V:\langle \mathbf {u} ,\mathbf {v} \rangle =0\ \ \forall \ \mathbf {u} \in W\}}">

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<annotation encoding="application/x-tex">{\displaystyle W^{\perp }=\{\mathbf {v} \in V:\langle \mathbf {u} ,\mathbf {v} \rangle =0\ \ \forall \ \mathbf {u} \in W\}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle W^{\perp }=\{\mathbf {v} \in V:\mathbf {\tilde {A}} y=\mathbf {v} ,\ y\in \mathbb {R} ^{3}\},}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {\tilde {A}} ={\begin{pmatrix}-2&-3&-5\\-6&-9&-3\\1&0&0\\0&1&0\\0&0&1\end{pmatrix}}.}</annotation>

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</p><p>The fact that every <a href="Row_and_column_vectors" title="Row and column vectors">column vector</a> 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 \mathbf {A} }">

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

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</math></span><img src="./0795cc96c75d81520a120482662b90f024c9a1a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }" loading="lazy"></span> is orthogonal to every column vector 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 \mathbf {\tilde {A}} }">

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

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</math></span><img src="./f9a2e102aca44f570cc715ee91129b8155a4b573.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.843ex;" alt="{\displaystyle \mathbf {\tilde {A}} }" loading="lazy"></span> can be checked by direct computation. The fact that the spans of these vectors are orthogonal then follows by bilinearity of the dot product. Finally, the fact that these spaces are orthogonal complements follows from the dimension relationships given below.

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<div class="mw-heading mw-heading2"><h2 id="General_bilinear_forms">General bilinear forms</h2></div>

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

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</math></span><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"></span> be a vector space over a <a href="Field_(mathematics)" title="Field (mathematics)">field</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 \mathbb {F} }">

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

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</math></span><img src="./573f72afae7df709959ab1a58cd643743466a187.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {F} }" loading="lazy"></span> equipped with a <a href="Bilinear_form" title="Bilinear form">bilinear form</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 B.}">

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

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</math></span><img src="./0eccf5bca7cdc1fa4439af2d31831db6bde00473.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.411ex; height:2.176ex;" alt="{\displaystyle B.}" loading="lazy"></span> We define <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 \mathbf {u} }">

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</math></span><img src="./261e20fe101de02a771021d9d4466c0ad3e352d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {u} }" loading="lazy"></span> to be left-orthogonal to <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 \mathbf {v} }">

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</math></span><img src="./35c1866e359fbfd2e0f606c725ba5cc37a5195d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }" loading="lazy"></span>, and <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 \mathbf {v} }">

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</math></span><img src="./35c1866e359fbfd2e0f606c725ba5cc37a5195d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }" loading="lazy"></span> to be right-orthogonal to <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 \mathbf {u} }">

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

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</math></span><img src="./261e20fe101de02a771021d9d4466c0ad3e352d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {u} }" loading="lazy"></span>, when <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 B(\mathbf {u} ,\mathbf {v} )=0.}">

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<annotation encoding="application/x-tex">{\displaystyle B(\mathbf {u} ,\mathbf {v} )=0.}</annotation>

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</math></span><img src="./a01ec6f5851664a615e0f3c6f2a62b24a43fff61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.411ex; height:2.843ex;" alt="{\displaystyle B(\mathbf {u} ,\mathbf {v} )=0.}" loading="lazy"></span> For a subset <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 W}">

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

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</math></span><img src="./ace9595e3ce66fdec7e9d30202626accd676b11e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.434ex; height:2.509ex;" alt="{\displaystyle V,}" loading="lazy"></span> define the left-orthogonal complement <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 W^{\perp }}">

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

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</math></span><img src="./dda1dcc2d841bdc9aefcf9c90799310a56fcad33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.019ex; height:2.676ex;" alt="{\displaystyle W^{\perp }}" loading="lazy"></span> to be

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<annotation encoding="application/x-tex">{\displaystyle W^{\perp }=\left\{\mathbf {x} \in V:B(\mathbf {x} ,\mathbf {y} )=0\ \ \forall \ \mathbf {y} \in W\right\}.}</annotation>

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</p><p>There is a corresponding definition of the right-orthogonal complement. For a <a href="Reflexive_bilinear_form" class="mw-redirect" title="Reflexive bilinear form">reflexive bilinear form</a>, where <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 B(\mathbf {u} ,\mathbf {v} )=0\implies B(\mathbf {v} ,\mathbf {u} )=0\ \ \forall \ \mathbf {u} ,\mathbf {v} \in V}">

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<annotation encoding="application/x-tex">{\displaystyle B(\mathbf {u} ,\mathbf {v} )=0\implies B(\mathbf {v} ,\mathbf {u} )=0\ \ \forall \ \mathbf {u} ,\mathbf {v} \in V}</annotation>

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</math></span><img src="./1e5c650aae6a0ecd6a598dbe79b1a26db5bfc27c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.508ex; height:2.843ex;" alt="{\displaystyle B(\mathbf {u} ,\mathbf {v} )=0\implies B(\mathbf {v} ,\mathbf {u} )=0\ \ \forall \ \mathbf {u} ,\mathbf {v} \in V}" loading="lazy"></span>, the left and right complements coincide. This will be the case if <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 B}">

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

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</math></span><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"></span> is a <a href="Symmetric_bilinear_form" title="Symmetric bilinear form">symmetric</a> or an <a href="Bilinear_form#Symmetric,_skew-symmetric_and_alternating_forms" title="Bilinear form">alternating form</a>.

</p><p>The definition extends to a bilinear form on a <a href="Free_module" title="Free module">free module</a> over a <a href="Commutative_ring" title="Commutative ring">commutative ring</a>, and to a <a href="Sesquilinear_form" title="Sesquilinear form">sesquilinear form</a> extended to include any free module over a commutative ring with <a href="Conjugate_element_(field_theory)" title="Conjugate element (field theory)">conjugation</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>

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

<ul><li>An orthogonal complement is a subspace of <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 V}">

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

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</math></span><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"></span>;</li>

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

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</math></span><img src="./894a17f6bbae8bb911fb08785e051b353e832ea1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.852ex; height:2.343ex;" alt="{\displaystyle X\subseteq Y}" loading="lazy"></span> then <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^{\perp }\supseteq Y^{\perp }}">

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

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</math></span><img src="./14e7cbe479dad27b18614f1d0d770d025ac0f13b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.017ex; height:2.843ex;" alt="{\displaystyle X^{\perp }\supseteq Y^{\perp }}" loading="lazy"></span>;</li>

<li>The <a href="Radical_of_a_quadratic_space" class="mw-redirect" title="Radical of a quadratic space">radical</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 V^{\perp }}">

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

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</math></span><img src="./2640c3a2d5c5b7fdc7afc31137bac2c6e4cef4bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.428ex; height:2.676ex;" alt="{\displaystyle V^{\perp }}" loading="lazy"></span> of <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 V}">

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

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</math></span><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"></span> is a subspace of every orthogonal complement;</li>

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<annotation encoding="application/x-tex">{\displaystyle W\subseteq (W^{\perp })^{\perp }}</annotation>

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</math></span><img src="./63c8f0205e8090aa7abc332ab963274cfa5c0e7d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.872ex; height:3.176ex;" alt="{\displaystyle W\subseteq (W^{\perp })^{\perp }}" loading="lazy"></span>;</li>

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

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</math></span><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"></span> is <a href="Non-degenerate" class="mw-redirect" title="Non-degenerate">non-degenerate</a> and <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 V}">

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

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</math></span><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"></span> is finite-dimensional, then <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 \dim(W)+\dim(W^{\perp })=\dim(V)}">

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<annotation encoding="application/x-tex">{\displaystyle \dim(W)+\dim(W^{\perp })=\dim(V)}</annotation>

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</math></span><img src="./22f54cd70267707ecf12ae1827664dbd2d3f0f8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.234ex; height:3.176ex;" alt="{\displaystyle \dim(W)+\dim(W^{\perp })=\dim(V)}" loading="lazy"></span>.</li>

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<annotation encoding="application/x-tex">{\displaystyle L_{1},\ldots ,L_{r}}</annotation>

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</math></span><img src="./62cbbdf14d0f03566c5ebb52a10c48bf59403f4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.372ex; height:2.509ex;" alt="{\displaystyle L_{1},\ldots ,L_{r}}" loading="lazy"></span> are subspaces of a finite-dimensional 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 V}">

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

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</math></span><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"></span> and <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_{*}=L_{1}\cap \cdots \cap L_{r},}">

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<annotation encoding="application/x-tex">{\displaystyle L_{*}=L_{1}\cap \cdots \cap L_{r},}</annotation>

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</math></span><img src="./afd5265ad0ed20956579d55491c05d3fbfe9011b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.464ex; height:2.509ex;" alt="{\displaystyle L_{*}=L_{1}\cap \cdots \cap L_{r},}" loading="lazy"></span> then <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_{*}^{\perp }=L_{1}^{\perp }+\cdots +L_{r}^{\perp }}">

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<annotation encoding="application/x-tex">{\displaystyle L_{*}^{\perp }=L_{1}^{\perp }+\cdots +L_{r}^{\perp }}</annotation>

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</math></span><img src="./f665afd1455bb39af7aa6850cbb7fbd2c4dc17a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.783ex; height:3.176ex;" alt="{\displaystyle L_{*}^{\perp }=L_{1}^{\perp }+\cdots +L_{r}^{\perp }}" loading="lazy"></span>.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Inner_product_spaces">Inner product spaces</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="Orthogonal_projection" class="mw-redirect" title="Orthogonal projection">Orthogonal projection</a></div>

<p>This section considers orthogonal complements in an <a href="Inner_product_space" title="Inner product space">inner product space</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 H}">

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

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</math></span><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.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 H}" loading="lazy"></span>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p><p>Two vectors <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 \mathbf {x} }">

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

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</math></span><img src="./32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }" loading="lazy"></span> and <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 \mathbf {y} }">

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</math></span><img src="./bb25a040b592282dc2a254c3117e792c3c81161f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.411ex; height:2.009ex;" alt="{\displaystyle \mathbf {y} }" loading="lazy"></span> are called <em><a href="Orthogonal_vectors_(inner_product_space)" class="mw-redirect" title="Orthogonal vectors (inner product space)">orthogonal</a></em> if <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 \langle \mathbf {x} ,\mathbf {y} \rangle =0}">

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<annotation encoding="application/x-tex">{\displaystyle \langle \mathbf {x} ,\mathbf {y} \rangle =0}</annotation>

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</math></span><img src="./91c94233efe3dbc09548f4716e5c44438c7ea8a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.926ex; height:2.843ex;" alt="{\displaystyle \langle \mathbf {x} ,\mathbf {y} \rangle =0}" loading="lazy"></span>, which happens <a href="If_and_only_if" title="If and only if">if and only if</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 \|\mathbf {x} \|\leq \|\mathbf {x} +s\mathbf {y} \|\ \forall }">

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<annotation encoding="application/x-tex">{\displaystyle \|\mathbf {x} \|\leq \|\mathbf {x} +s\mathbf {y} \|\ \forall }</annotation>

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</math></span><img src="./99392a391f80d675756f7f199af4c98d78c4df98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.785ex; height:2.843ex;" alt="{\displaystyle \|\mathbf {x} \|\leq \|\mathbf {x} +s\mathbf {y} \|\ \forall }" loading="lazy"></span> scalars <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 s}">

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

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</math></span><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"></span>.<sup id="cite_ref-FOOTNOTERudin1991306–312_3-0" class="reference"><a href="#cite_note-FOOTNOTERudin1991306–312-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p><p>If <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 C}">

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

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</math></span><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"></span> is any subset of an inner product 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 H}">

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

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</math></span><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.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 H}" loading="lazy"></span> then its <em><style data-mw-deduplicate="TemplateStyles:r1238216509">

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</style><span class="vanchor"><span class="vanchor-text">orthogonal complement</span></span> 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 H}">

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

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</math></span><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.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 H}" loading="lazy"></span></em> is the vector subspace

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}C^{\perp }:&=\{\mathbf {x} \in H:\langle \mathbf {x} ,\mathbf {c} \rangle =0\ \ \forall \ \mathbf {c} \in C\}\\&=\{\mathbf {x} \in H:\langle \mathbf {c} ,\mathbf {x} \rangle =0\ \ \forall \ \mathbf {c} \in C\}\end{aligned}}}</annotation>

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which is always a <a href="Closed_set" title="Closed set">closed subset</a> (hence, a closed vector subspace) of <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 H}">

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

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</math></span><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.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 H}" loading="lazy"></span><sup id="cite_ref-FOOTNOTERudin1991306–312_3-1" class="reference"><a href="#cite_note-FOOTNOTERudin1991306–312-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>proof 1<span class="cite-bracket">]</span></a></sup> that satisfies:

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<annotation encoding="application/x-tex">{\displaystyle C^{\bot }=\left(\operatorname {cl} _{H}\left(\operatorname {span} C\right)\right)^{\bot }}</annotation>

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</math></span><img src="./bbb8bd5116bedd7c4d08d8d67aaeb407aa695051.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.724ex; height:3.343ex;" alt="{\displaystyle C^{\bot }=\left(\operatorname {cl} _{H}\left(\operatorname {span} C\right)\right)^{\bot }}" loading="lazy"></span>;</li>

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<annotation encoding="application/x-tex">{\displaystyle C^{\bot }\cap \operatorname {cl} _{H}\left(\operatorname {span} C\right)=\{0\}}</annotation>

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</math></span><img src="./f5a7bdd8aeb5861af9f594db275184dfbd3d0727.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.474ex; height:3.176ex;" alt="{\displaystyle C^{\bot }\cap \operatorname {cl} _{H}\left(\operatorname {span} C\right)=\{0\}}" loading="lazy"></span>;</li>

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<annotation encoding="application/x-tex">{\displaystyle C^{\bot }\cap \left(\operatorname {span} C\right)=\{0\}}</annotation>

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</math></span><img src="./bb07f0394ed73a18033f8806c0c13761ba6b3c8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.104ex; height:3.176ex;" alt="{\displaystyle C^{\bot }\cap \left(\operatorname {span} C\right)=\{0\}}" loading="lazy"></span>;</li>

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<annotation encoding="application/x-tex">{\displaystyle C\subseteq \left(C^{\bot }\right)^{\bot }}</annotation>

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</math></span><img src="./613cc3d4e9925305202999c242ce9cf6068237fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.814ex; height:3.843ex;" alt="{\displaystyle C\subseteq \left(C^{\bot }\right)^{\bot }}" loading="lazy"></span>;</li>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} _{H}\left(\operatorname {span} C\right)\subseteq \left(C^{\bot }\right)^{\bot }}</annotation>

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</math></span><img src="./8b367222585a346f7c6de9c587eec5ae21f267c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.045ex; height:3.843ex;" alt="{\displaystyle \operatorname {cl} _{H}\left(\operatorname {span} C\right)\subseteq \left(C^{\bot }\right)^{\bot }}" loading="lazy"></span>.</li></ul>

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

</semantics>

</math></span><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"></span> is a vector subspace of an inner product 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 H}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.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 H}" loading="lazy"></span> then

<msup>

</mrow>

</msup>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C^{\bot }=\left\{\mathbf {x} \in H:\|\mathbf {x} \|\leq \|\mathbf {x} +\mathbf {c} \|\ \ \forall \ \mathbf {c} \in C\right\}.}</annotation>

</semantics>

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

If <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 C}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"></span> is a closed vector subspace of a Hilbert 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 H}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.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 H}" loading="lazy"></span> then<sup id="cite_ref-FOOTNOTERudin1991306–312_3-2" class="reference"><a href="#cite_note-FOOTNOTERudin1991306–312-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

<msup>

</mrow>

</msup>

</mrow>

<msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H=C\oplus C^{\bot }\qquad {\text{ and }}\qquad \left(C^{\bot }\right)^{\bot }=C}</annotation>

</semantics>

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

where <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 H=C\oplus C^{\bot }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H=C\oplus C^{\bot }}</annotation>

</semantics>

</math></span><img src="./b05e5439f2d35acba2b360d517afa04248f3aace.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.077ex; height:2.843ex;" alt="{\displaystyle H=C\oplus C^{\bot }}" loading="lazy"></span> is called the <em><span class="vanchor"><span class="vanchor-text">orthogonal decomposition</span></span></em> of <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 H}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.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 H}" loading="lazy"></span> into <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 C}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"></span> and <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 C^{\bot }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./e9ba6bddf4b4b174c7ce96ac011aae08f4990689.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.309ex; height:2.676ex;" alt="{\displaystyle C^{\bot }}" loading="lazy"></span> and it indicates that <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 C}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"></span> is a <a href="Complemented_subspace" title="Complemented subspace">complemented subspace</a> of <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 H}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.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 H}" loading="lazy"></span> with complement <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 C^{\bot }.}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./60f41eaa997b3c9e1822ca1647ddb7cef36aaf47.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.955ex; height:2.676ex;" alt="{\displaystyle C^{\bot }.}" loading="lazy"></span>

</p>

<div class="mw-heading mw-heading3"><h3 id="Properties_2">Properties</h3></div>

<p>The orthogonal complement is always closed in the <a href="Metric_space" title="Metric space">metric topology</a>. In finite-dimensional spaces, that is merely an instance of the fact that all subspaces of a vector space are closed. In infinite-dimensional <a href="Hilbert_space" title="Hilbert space">Hilbert spaces</a>, some subspaces are not closed, but all orthogonal complements are closed. If <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 W}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"></span> is a vector subspace of a <a href="Hilbert_space" title="Hilbert space">Hilbert space</a> the orthogonal complement of the orthogonal complement of <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 W}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"></span> is the <a href="Closure_(topology)" title="Closure (topology)">closure</a> of <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 W,}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./e3e52f6de64b3b96ce1db5651329f808cca7d3b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.082ex; height:2.509ex;" alt="{\displaystyle W,}" loading="lazy"></span> that is,

<msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msup>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(W^{\bot }\right)^{\bot }={\overline {W}}.}</annotation>

</semantics>

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

</p><p>Some other useful properties that always hold are the following. Let <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 H}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.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 H}" loading="lazy"></span> be a Hilbert space and let <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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> and <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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> be linear subspaces. Then:

</p>

<msup>

</mrow>

</msup>

<msup>

<mover>

</mover>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./9d7d0cfc9bc847ea7364457effcfe42e5289a79d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.248ex; height:3.343ex;" alt="{\displaystyle X^{\bot }={\overline {X}}^{\bot }}" loading="lazy"></span>;</li>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./abf3e597919d6ae699180ce66b51edbe14f3546a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.852ex; height:2.343ex;" alt="{\displaystyle Y\subseteq X}" loading="lazy"></span> then <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^{\bot }\subseteq Y^{\bot }}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./13d394cfc2065c1f540a17e9cf83590f42bd0492.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.017ex; height:2.843ex;" alt="{\displaystyle X^{\bot }\subseteq Y^{\bot }}" loading="lazy"></span>;</li>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./4481bc9378af70ffbee800101b6891e4890f8fe0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.656ex; height:3.176ex;" alt="{\displaystyle X\cap X^{\bot }=\{0\}}" loading="lazy"></span>;</li>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\subseteq (X^{\bot })^{\bot }}</annotation>

</semantics>

</math></span><img src="./c6bf76ea1b32632c5d4885be9b332e4725a9a820.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.906ex; height:3.176ex;" alt="{\displaystyle X\subseteq (X^{\bot })^{\bot }}" loading="lazy"></span>;</li>

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> is a closed linear subspace of <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 H}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.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 H}" loading="lazy"></span> then <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^{\bot })^{\bot }=X}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./8c44bd4e489ecb369069598b87a842c1fc4d9230.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.906ex; height:3.176ex;" alt="{\displaystyle (X^{\bot })^{\bot }=X}" loading="lazy"></span>;</li>

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> is a closed linear subspace of <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 H}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.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 H}" loading="lazy"></span> then <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 H=X\oplus X^{\bot },}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./d612acee33b715644bb4b901890323a9c28be3a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.137ex; height:3.009ex;" alt="{\displaystyle H=X\oplus X^{\bot },}" loading="lazy"></span> the (inner) <a href="Direct_sum" title="Direct sum">direct sum</a>.</li></ul>

<p>The orthogonal complement generalizes to the <a href="Annihilator_(ring_theory)" title="Annihilator (ring theory)">annihilator</a>, and gives a <a href="Galois_connection" title="Galois connection">Galois connection</a> on subsets of the inner product space, with associated <a href="Closure_operator" title="Closure operator">closure operator</a> the topological closure of the span.

</p>

<div class="mw-heading mw-heading3"><h3 id="Finite_dimensions">Finite dimensions</h3></div>

<p>For a finite-dimensional inner product space of dimension <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 n}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>, the orthogonal complement of 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 k}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span>-dimensional subspace is an <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 (n-k)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./66e9f25b19fdb4fc155f89d4f926e9d9e4e440da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.256ex; height:2.843ex;" alt="{\displaystyle (n-k)}" loading="lazy"></span>-dimensional subspace, and the double orthogonal complement is the original subspace:

<msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(W^{\bot }\right)^{\bot }=W.}</annotation>

</semantics>

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

</p><p>If <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 \mathbf {A} \in \mathbb {M} _{mn}}">

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} \in \mathbb {M} _{mn}}</annotation>

</semantics>

</math></span><img src="./7fadb6b1ce662efe972189f79307009fac7f2da9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.715ex; height:2.509ex;" alt="{\displaystyle \mathbf {A} \in \mathbb {M} _{mn}}" loading="lazy"></span>, where <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 {R}}(\mathbf {A} )}">

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./300e9008206d88f4d4a331f1ec3dd162e5f4dfbc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.8ex; height:2.843ex;" alt="{\displaystyle {\mathcal {R}}(\mathbf {A} )}" loading="lazy"></span>, <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}}(\mathbf {A} )}">

</mrow>

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}(\mathbf {A} )}</annotation>

</semantics>

</math></span><img src="./0dc150cf6a7c52efc041c2d75fe66464202b86c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.068ex; height:2.843ex;" alt="{\displaystyle {\mathcal {C}}(\mathbf {A} )}" loading="lazy"></span>, and <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 {N}}(\mathbf {A} )}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}(\mathbf {A} )}</annotation>

</semantics>

</math></span><img src="./68917ae2bc237fa22ad17711c45ab91b48b349a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.062ex; width:6.165ex; height:3.009ex;" alt="{\displaystyle {\mathcal {N}}(\mathbf {A} )}" loading="lazy"></span> refer to the <a href="Row_space" class="mw-redirect" title="Row space">row space</a>, <a href="Column_space" class="mw-redirect" title="Column space">column space</a>, and <a href="Null_space" class="mw-redirect" title="Null space">null space</a> of <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 \mathbf {A} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./0795cc96c75d81520a120482662b90f024c9a1a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }" loading="lazy"></span> (respectively), then<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

<msup>

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<msup>

<mrow>

<mrow>

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<msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left({\mathcal {R}}(\mathbf {A} )\right)^{\bot }={\mathcal {N}}(\mathbf {A} )\qquad {\text{ and }}\qquad \left({\mathcal {C}}(\mathbf {A} )\right)^{\bot }={\mathcal {N}}(\mathbf {A} ^{\operatorname {T} }).}</annotation>

</semantics>

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

</p>

<div class="mw-heading mw-heading2"><h2 id="Banach_spaces">Banach spaces</h2></div>

<p>There is a natural analog of this notion in general <a href="Banach_space" title="Banach space">Banach spaces</a>. In this case one defines the orthogonal complement of <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 W}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"></span> to be a subspace of the <a href="Dual_space" title="Dual space">dual</a> of <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 V}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> defined similarly as the <a href="Dual_space#Annihilators" title="Dual space">annihilator</a>

<msup>

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<mrow>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle W^{\bot }=\left\{x\in V^{*}:\forall y\in W,x(y)=0\right\}.}</annotation>

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

</p><p>It is always a closed subspace of <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 V^{*}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./5910e6a94f4f7ee2ee85ceed9dacef3eff7a6242.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle V^{*}}" loading="lazy"></span>. There is also an analog of the double complement property. <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 W^{\perp \perp }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./74649e3cd1cb14d8895c2f9dca7e9a166eca792c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.297ex; height:2.676ex;" alt="{\displaystyle W^{\perp \perp }}" loading="lazy"></span> is now a subspace of <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 V^{**}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./94a8dd7436855307c1632d3a96c64ee542a08426.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.793ex; height:2.343ex;" alt="{\displaystyle V^{**}}" loading="lazy"></span>(which is not identical to <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 V}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span>). However, the <a href="Reflexive_space" title="Reflexive space">reflexive spaces</a> have a <a href="Natural_transformation" title="Natural transformation">natural</a> <a href="Isomorphism" title="Isomorphism">isomorphism</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 i}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span> between <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 V}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> and <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 V^{**}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./94a8dd7436855307c1632d3a96c64ee542a08426.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.793ex; height:2.343ex;" alt="{\displaystyle V^{**}}" loading="lazy"></span>. In this case we have <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i{\overline {W}}=W^{\perp \perp }.}">

<mover>

</mover>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i{\overline {W}}=W^{\perp \perp }.}</annotation>

</semantics>

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

</p><p>This is a rather straightforward consequence of the <a href="Hahn%E2%80%93Banach_theorem" title="Hahn–Banach theorem">Hahn–Banach theorem</a>.

</p>

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

<p>In <a href="Special_relativity" title="Special relativity">special relativity</a> the orthogonal complement is used to determine the <a href="World_line#Simultaneous_hyperplane" title="World line">simultaneous hyperplane</a> at a point of a <a href="World_line" title="World line">world line</a>. The bilinear form <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 \eta }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./e4d701857cf5fbec133eebaf94deadf722537f64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }" loading="lazy"></span> used in <a href="Minkowski_space" title="Minkowski space">Minkowski space</a> determines a <a href="Pseudo-Euclidean_space" title="Pseudo-Euclidean space">pseudo-Euclidean space</a> of events.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> The origin and all events on the <a href="Light_cone" title="Light cone">light cone</a> are self-orthogonal. When a <a href="Time" title="Time">time</a> event and a <a href="Space" title="Space">space</a> event evaluate to zero under the bilinear form, then they are <a href="Hyperbolic-orthogonal" class="mw-redirect" title="Hyperbolic-orthogonal">hyperbolic-orthogonal</a>. This terminology stems from the use of <a href="Conjugate_hyperbola" title="Conjugate hyperbola">conjugate hyperbolas</a> in the pseudo-Euclidean plane: <a href="Conjugate_diameters" title="Conjugate diameters">conjugate diameters</a> of these hyperbolas are hyperbolic-orthogonal.

</p>

<ul><li><a href="Complemented_lattice" title="Complemented lattice">Complemented lattice</a> – Bound lattice in which every element has a complement</li>

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

<li><a href="Hilbert_projection_theorem" title="Hilbert projection theorem">Hilbert projection theorem</a> – On closed convex subsets in Hilbert space</li>

<li><a href="Orthogonal_projection" class="mw-redirect" title="Orthogonal projection">Orthogonal projection</a> – Idempotent linear transformation from a vector space to itself<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li></ul>

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

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

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</math></span><img src="./391d6799ae2df6d2c41503c66cca043a5ca607a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.673ex; height:2.176ex;" alt="{\displaystyle C=\varnothing }" loading="lazy"></span> then <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 C^{\bot }=H,}">

<msup>

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

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</math></span><img src="./f2393d724ee96044fc2fffeb9e3fe9f813c2cd45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.118ex; height:3.009ex;" alt="{\displaystyle C^{\bot }=H,}" loading="lazy"></span> which is closed 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 H}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.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 H}" loading="lazy"></span> so assume <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 C\neq \varnothing .}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./ce440c411869e1f9a64c6fdbf64cafc7ee801bbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.32ex; height:2.676ex;" alt="{\displaystyle C\neq \varnothing .}" loading="lazy"></span> Let <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 P:=\prod _{c\in C}\mathbb {F} }">

</mrow>

</munder>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle P:=\prod _{c\in C}\mathbb {F} }</annotation>

</semantics>

</math></span><img src="./d6c0ed4fb8f931dc7cd44a5ea59b9491a6708a3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.781ex; height:3.009ex;" alt="{\textstyle P:=\prod _{c\in C}\mathbb {F} }" loading="lazy"></span> where <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 {F} }">

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</mstyle>

</mrow>

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

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</math></span><img src="./573f72afae7df709959ab1a58cd643743466a187.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {F} }" loading="lazy"></span> is the underlying scalar field of <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 H}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.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 H}" loading="lazy"></span> and define <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:H\to P}">

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<annotation encoding="application/x-tex">{\displaystyle L:H\to P}</annotation>

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</math></span><img src="./11b886e2bf4faca7a51e8fa6b8b4d0c4c220c293.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.943ex; height:2.176ex;" alt="{\displaystyle L:H\to P}" loading="lazy"></span> by <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(h):=\left(\langle h,c\rangle \right)_{c\in C},}">

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<annotation encoding="application/x-tex">{\displaystyle L(h):=\left(\langle h,c\rangle \right)_{c\in C},}</annotation>

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</math></span><img src="./f0f65ae459f1b5066c5782f0b78d7049438b7feb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.411ex; height:3.009ex;" alt="{\displaystyle L(h):=\left(\langle h,c\rangle \right)_{c\in C},}" loading="lazy"></span> which is continuous because this is true of each of its coordinates <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 h\mapsto \langle h,c\rangle .}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h\mapsto \langle h,c\rangle .}</annotation>

</semantics>

</math></span><img src="./e6926e795477171a66ba8fe3fed0e837755024be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.789ex; height:2.843ex;" alt="{\displaystyle h\mapsto \langle h,c\rangle .}" loading="lazy"></span> Then <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 C^{\bot }=L^{-1}(0)=L^{-1}\left(\{0\}\right)}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

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</msup>

<mrow>

<mrow>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle C^{\bot }=L^{-1}(0)=L^{-1}\left(\{0\}\right)}</annotation>

</semantics>

</math></span><img src="./0d4469752211bba2e40c621bcb7309d8c0f79a03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.992ex; height:3.176ex;" alt="{\displaystyle C^{\bot }=L^{-1}(0)=L^{-1}\left(\{0\}\right)}" loading="lazy"></span> is closed 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 H}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.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 H}" loading="lazy"></span> because <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 \{0\}}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./0ff0df9ef65c0572eb676580ce1c02b8ec40f694.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle \{0\}}" loading="lazy"></span> is closed 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 P}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span> and <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:H\to P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L:H\to P}</annotation>

</semantics>

</math></span><img src="./11b886e2bf4faca7a51e8fa6b8b4d0c4c220c293.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.943ex; height:2.176ex;" alt="{\displaystyle L:H\to P}" loading="lazy"></span> is continuous. If <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 \langle \,\cdot \,,\,\cdot \,\rangle }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./2027b7f5f03c87f88b943b87e1062cc837e60dd6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.685ex; height:2.843ex;" alt="{\displaystyle \langle \,\cdot \,,\,\cdot \,\rangle }" loading="lazy"></span> is linear in its first (respectively, its second) coordinate then <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:H\to P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L:H\to P}</annotation>

</semantics>

</math></span><img src="./11b886e2bf4faca7a51e8fa6b8b4d0c4c220c293.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.943ex; height:2.176ex;" alt="{\displaystyle L:H\to P}" loading="lazy"></span> is a <a href="Linear_map" title="Linear map">linear map</a> (resp. an <a href="Antilinear_map" title="Antilinear map">antilinear map</a>); either way, its kernel <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 \operatorname {ker} L=L^{-1}(0)=C^{\bot }}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./e46f7ade62e64c97c39b0ac87147ce9f371c683a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.534ex; height:3.176ex;" alt="{\displaystyle \operatorname {ker} L=L^{-1}(0)=C^{\bot }}" loading="lazy"></span> is a vector subspace of <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 H.}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./8933ae7244305ae7824aa18e077d1cf946e2ee9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.71ex; height:2.176ex;" alt="{\displaystyle H.}" loading="lazy"></span> <a href="Q.E.D." title="Q.E.D.">Q.E.D.</a></span>

</li>

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

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

<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Adkins & Weintraub (1992) p.359</span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Adkins&Weintraub (1992) p.272</span>

</li>

<li id="cite_note-FOOTNOTERudin1991306–312-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTERudin1991306–312_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTERudin1991306–312_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-FOOTNOTERudin1991306–312_3-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFRudin1991">Rudin 1991</a>, pp. 306–312.</span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://www.mathwizurd.com/linalg/2018/12/10/orthogonal-complement">"Orthogonal Complement"</a></span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a href="G._D._Birkhoff" class="mw-redirect" title="G. D. Birkhoff">G. D. Birkhoff</a> (1923) <i>Relativity and Modern Physics</i>, pages 62,63, <a href="Harvard_University_Press" title="Harvard University Press">Harvard University Press</a></span>

</li>

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

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

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

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</style><cite id="CITEREFAdkinsWeintraub1992" class="citation cs2">Adkins, William A.; Weintraub, Steven H. (1992), <i>Algebra: An Approach via Module Theory</i>, <a href="Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>, vol. 136, <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-97839-9</bdi>, <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0768.00003">0768.00003</a></cite></li>

<li><cite id="CITEREFHalmos1974" class="citation cs2"><a href="Paul_R._Halmos" class="mw-redirect" title="Paul R. Halmos">Halmos, Paul R.</a> (1974), <i>Finite-dimensional vector spaces</i>, <a href="Undergraduate_Texts_in_Mathematics" title="Undergraduate Texts in Mathematics">Undergraduate Texts in Mathematics</a>, Berlin, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-90093-3</bdi>, <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0288.15002">0288.15002</a></cite></li>

<li><cite id="CITEREFMilnorHusemoller1973" class="citation cs2"><a href="John_Milnor" title="John Milnor">Milnor, J.</a>; Husemoller, D. (1973), <i>Symmetric Bilinear Forms</i>, <a href="Ergebnisse_der_Mathematik_und_ihrer_Grenzgebiete" title="Ergebnisse der Mathematik und ihrer Grenzgebiete">Ergebnisse der Mathematik und ihrer Grenzgebiete</a>, vol. 73, <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-06009-X</bdi>, <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0292.10016">0292.10016</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"><i>Functional Analysis</i></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></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li>Orthogonal <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=rnNxUekd3B4&t=645s">complement; Minute 9.00 in the Youtube Video</a></li>

<li><a rel="nofollow" class="external text" href="http://www.khanacademy.org/math/linear-algebra/alternate-bases/othogonal-complements/v/linear-algebra-orthogonal-complements">Instructional video describing orthogonal complements (Khan Academy)</a></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><span class="noviewer" typeof="mw:File"><span title="Category"></span></span> Category</li></ul>

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

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Hilbert_spaces69" 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="Hilbert_spaces69" style="font-size:114%;margin:0 4em"><a href="Hilbert_space" title="Hilbert space">Hilbert 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="Hermitian_adjoint" title="Hermitian adjoint">Adjoint</a></li>

<li><a href="Inner_product_space" title="Inner product space">Inner product</a> and <a href="L-semi-inner_product" title="L-semi-inner product">L-semi-inner product</a></li>

<li><a href="Hilbert_space" title="Hilbert space">Hilbert space</a> and <a href="Prehilbert_space" class="mw-redirect" title="Prehilbert space">Prehilbert space</a></li>

<li><a href="Orthonormal_basis" title="Orthonormal basis">Orthonormal basis</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="Bessel's_inequality" title="Bessel's inequality">Bessel's inequality</a></li>

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

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

<ul><li><a href="Hilbert_projection_theorem" title="Hilbert projection theorem">Hilbert projection theorem</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> (<a href="Parallelogram_law#The_parallelogram_law_in_inner_product_spaces" title="Parallelogram law">Parallelogram law</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-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Compact_operator_on_Hilbert_space" title="Compact operator on Hilbert space">Compact operator on Hilbert space</a></li>

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

<li><a href="Sesquilinear_form#Hermitian_form" title="Sesquilinear form">Hermitian form</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="Self-adjoint_operator" title="Self-adjoint operator">Self-adjoint</a></li>

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

<li><a href="Trace_class" title="Trace class">Trace class</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%">Examples</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="Distribution_(mathematics)" title="Distribution (mathematics)"><i>C</i><sup><i>n</i></sup>(<i>K</i>) with <i>K</i> compact & <i>n</i><∞</a></li>

<li><a href="Segal%E2%80%93Bargmann_space" title="Segal–Bargmann space">Segal–Bargmann <i>F</i></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 2025-07-12" href="https://en.wikipedia.org/wiki/?title=Orthogonal_complement&oldid=1300112309">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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