Editing Orthogonal complement (section)

== Inner product spaces ==
{{See also|Orthogonal projection}}

This section considers orthogonal complements in an [[inner product space]] <math>H</math>.<ref>Adkins&Weintraub (1992) p.272</ref>

Two vectors <math>\mathbf{x}</math> and <math>\mathbf{y}</math> are called {{em|[[Orthogonal vectors (inner product space)|orthogonal]]}} if <math>\langle \mathbf{x}, \mathbf{y} \rangle = 0</math>, which happens [[if and only if]] <math>\| \mathbf{x} \| \le \| \mathbf{x} + s\mathbf{y} \| \ \forall</math> scalars <math>s</math>.{{sfn|Rudin|1991|pp=306-312}}

If <math>C</math> is any subset of an inner product space <math>H</math> then its {{em|{{visible anchor|orthogonal complement}} in <math>H</math>}} is the vector subspace
<math display="block">\begin{align}
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{align}</math>
which is always a closed subset (hence, a closed vector subspace) of <math>H</math>{{sfn|Rudin|1991|pp=306-312}}<ref group="proof">If <math>C = \varnothing</math> then <math>C^{\bot} = H,</math> which is closed in <math>H</math> so assume <math>C \neq \varnothing.</math> Let <math display="inline">P := \prod_{c \in C} \mathbb{F}</math> where <math>\mathbb{F}</math> is the underlying scalar field of <math>H</math> and define <math>L : H \to P</math> by <math>L(h) := \left(\langle h, c \rangle\right)_{c \in C},</math> which is continuous because this is true of each of its coordinates <math>h \mapsto \langle h, c \rangle.</math> Then <math>C^{\bot} = L^{-1}(0) = L^{-1}\left(\{ 0 \}\right)</math> is closed in <math>H</math> because <math>\{ 0 \}</math> is closed in <math>P</math> and <math>L : H \to P</math> is continuous. If <math>\langle \,\cdot\,, \,\cdot\, \rangle</math> is linear in its first (respectively, its second) coordinate then <math>L : H \to P</math> is a [[linear map]] (resp. an [[antilinear map]]); either way, its kernel <math>\operatorname{ker} L = L^{-1}(0) = C^{\bot}</math> is a vector subspace of <math>H.</math> [[Q.E.D.]]</ref> that satisfies:
* <math>C^{\bot} = \left(\operatorname{cl}_H \left(\operatorname{span} C\right)\right)^{\bot}</math>;
* <math>C^{\bot} \cap \operatorname{cl}_H \left(\operatorname{span} C\right) = \{ 0 \}</math>;
* <math>C^{\bot} \cap  \left(\operatorname{span} C\right) = \{ 0 \}</math>;
* <math>C \subseteq \left(C^{\bot}\right)^{\bot}</math>;
* <math>\operatorname{cl}_H \left(\operatorname{span} C\right) \subseteq \left(C^{\bot}\right)^{\bot}</math>.

If <math>C</math> is a vector subspace of an inner product space <math>H</math> then 
<math display="block">C^{\bot} = \left\{\mathbf{x} \in H : \|\mathbf{x}\| \leq \|\mathbf{x} + \mathbf{c}\| \ \ \forall \ \mathbf{c} \in C \right\}.</math> 
If <math>C</math> is a closed vector subspace of a Hilbert space <math>H</math> then{{sfn|Rudin|1991|pp=306-312}} 
<math display="block">H = C \oplus C^{\bot} \qquad \text{ and } \qquad \left(C^{\bot}\right)^{\bot} = C</math>
where <math>H = C \oplus C^{\bot}</math> is called the {{em|{{visible anchor|orthogonal decomposition}}}} of <math>H</math> into <math>C</math> and <math>C^{\bot}</math> and it indicates that <math>C</math> is a [[complemented subspace]] of <math>H</math> with complement <math>C^{\bot}.</math>

=== Properties ===

The orthogonal complement is always closed in the metric topology.  In finite-dimensional spaces, that is merely an instance of the fact that all subspaces of a vector space are closed.  In infinite-dimensional [[Hilbert space]]s, some subspaces are not closed, but all orthogonal complements are closed. If <math>W</math> is a vector subspace of a [[Hilbert space]] the orthogonal complement of the orthogonal complement of <math>W</math> is the [[Closure (topology)|closure]] of <math>W,</math> that is,
<math display="block">\left(W^\bot\right)^\bot = \overline W.</math>

Some other useful properties that always hold are the following. Let <math>H</math> be a Hilbert space and let <math>X</math> and <math>Y</math> be linear subspaces. Then:
* <math>X^\bot = \overline{X}^{\bot}</math>;
* if <math>Y \subseteq X</math> then <math>X^\bot \subseteq Y^\bot</math>;
* <math>X \cap X^\bot = \{ 0 \}</math>;
* <math>X \subseteq (X^\bot)^\bot</math>;
* if <math>X</math> is a closed linear subspace of <math>H</math> then <math>(X^\bot)^\bot = X</math>;
* if <math>X</math> is a closed linear subspace of <math>H</math> then <math>H = X \oplus X^\bot,</math> the (inner) [[direct sum]].

The orthogonal complement generalizes to the [[Annihilator (ring theory)|annihilator]], and gives a [[Galois connection]] on subsets of the inner product space, with associated [[closure operator]] the topological closure of the span.

=== Finite dimensions ===

For a finite-dimensional inner product space of dimension <math>n</math>, the orthogonal complement of a <math>k</math>-dimensional subspace is an <math>(n-k)</math>-dimensional subspace, and the double orthogonal complement is the original subspace:
<math display="block">\left(W^{\bot}\right)^{\bot} = W.</math>

If <math>\mathbf{A} \in \mathbb{M}_{mn}</math>, where <math>\mathcal{R}(\mathbf{A})</math>, <math>\mathcal{C} (\mathbf{A})</math>, and <math>\mathcal{N} (\mathbf{A})</math> refer to the [[row space]], [[column space]], and [[null space]] of <math>\mathbf{A}</math> (respectively), then<ref>[https://www.mathwizurd.com/linalg/2018/12/10/orthogonal-complement "Orthogonal Complement"]</ref>
<math display="block">\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}}).</math>