Editing Inner product space (section)

===Orthogonality===

{{glossary}}
{{term|[[Orthogonality (mathematics)|Orthogonality]]}}{{defn|
Two vectors <math>x</math> and <math>y</math> are said to be {{em|{{visible anchor|orthogonal|Orthogonal vectors}}}}, often written <math>x \perp y,</math> if their inner product is zero, that is, if <math>\langle x, y \rangle = 0.</math> <br>

This happens if and only if <math>\|x\| \leq \|x + s y\|</math> for all scalars <math>s,</math>{{sfn|Rudin|1991|pp=306-312}} and if and only if the real-valued function <math>f(s) := \|x + s y\|^2 - \|x\|^2</math> is non-negative. (This is a consequence of the fact that, if <math>y \neq 0</math> then the scalar <math>s_0 = - \tfrac{\overline{\langle x, y \rangle}}{\|y\|^2}</math> minimizes <math>f</math> with value <math>f\left(s_0\right) = - \tfrac{|\langle x, y \rangle|^2}{\|y\|^2},</math> which is always non positive).<br>

For a {{em|complex}} inner product space <math>H,</math> a linear operator <math>T : V \to V</math> is identically <math>0</math> if and only if <math>x \perp T x</math> for every <math>x \in V.</math>{{sfn|Rudin|1991|pp=306-312}} This is not true in general for real inner product spaces, as it is a consequence of conjugate symmetry being distinct from symmetry for complex inner products. A counterexample in a real inner product space is <math>T</math> a 90° rotation in <math>\mathbb{R}^2</math>, which maps every vector to an orthogonal vector but is not identically <math>0</math>.
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{{term|[[Orthogonal complement]]}}{{defn|The ''orthogonal complement'' of a subset <math>C \subseteq V</math> is the set <math>C^{\bot}</math> of the vectors that are orthogonal to all elements of {{mvar|C}}; that is, 
<math display=block>C^{\bot} := \{\,y \in V : \langle y, c \rangle = 0 \text{ for all } c \in C\,\}.</math> 
This set <math>C^{\bot}</math> is always a closed vector subspace of <math>V</math> and if the [[Closure (topology)|closure]] <math>\operatorname{cl}_V C</math> of <math>C</math> in <math>V</math> is a vector subspace then <math>\operatorname{cl}_V C = \left(C^{\bot}\right)^{\bot}.</math> 
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{{term|[[Pythagorean theorem]]}}{{defn|
If <math>x</math> and <math>y</math> are orthogonal, then
<math display=block>\|x\|^2 + \|y\|^2 = \|x + y\|^2.</math>

This may be proved by  expressing the squared norms in terms of the inner products, using  additivity for expanding the right-hand side of the equation.<br>

The name {{em|Pythagorean theorem}} arises from the geometric interpretation in [[Euclidean geometry]].
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{{term|[[Parseval's identity]]}}{{defn|
An [[Mathematical induction|induction]] on the Pythagorean theorem yields: if <math>x_1, \ldots, x_n</math> are pairwise orthogonal, then
<math display=block>\sum_{i=1}^n \|x_i\|^2 = \left\|\sum_{i=1}^n x_i\right\|^2.</math>
}}
{{anchor|Angle}}{{term|[[Angle]]}}{{defn|
When <math>\langle x, y \rangle</math> is a real number then the Cauchy–Schwarz inequality implies that <math display=inline>\frac{\langle x, y \rangle}{\|x\| \, \|y\|} \in [-1, 1],</math> and thus that 
<math display=block>\angle(x, y) = \arccos \frac{\langle x, y \rangle}{\|x\| \, \|y\|},</math>
is a real number. This allows defining the (non oriented) {{em|angle}} of two vectors in modern definitions of [[Euclidean geometry]] in terms of [[linear algebra]]. This is also used in [[data analysis]], under the name "[[cosine similarity]]", for comparing two vectors of data.

Furthermore, if <math>\langle x, y \rangle</math> is negative, the angle <math>\angle(x, y)</math> is larger than 90 degrees. This property is often used in computer graphics (e.g., in [[back-face culling]]) to analyze a direction without having to evaluate [[trigonometric functions]].}}
{{glossary end}}