Editing Inner product space (section)

==Basic results, terminology, and definitions==
===Norm properties {{anchor|Norm}}===<!-- This section is linked from [[Cauchy–Schwarz inequality]] -->

Every inner product space induces a [[Norm (mathematics)|norm]], called its {{em|{{visible anchor|canonical norm}}}}, that is defined by
<math display=block>\|x\| = \sqrt{\langle x, x \rangle}.</math> 
With this norm, every inner product space becomes a [[normed vector space]]. 

So, every general property of normed vector spaces applies to inner product spaces. <!-- In particular, an inner product space is a  [[metric space]], for the distance defined by
<math display=block>d(x, y) = \|y - x\|.</math> -->
In particular, one has the following properties:

{{glossary}}
{{term|[[Absolute homogeneity]]}}{{defn|
<math display=block>\|ax\| = |a| \, \|x\|</math>
for every <math>x \in V</math> and <math>a \in F</math>
(this results from <math>\langle ax, ax \rangle = a\overline a \langle x, x \rangle</math>).
}}
{{term|[[Triangle inequality]]}}{{defn|
<math display=block>\|x + y\| \leq \|x\| + \|y\|</math>
for <math>x, y\in V.</math>
These two properties show that one has indeed a norm.}}

{{term|[[Cauchy–Schwarz inequality]]}}{{defn|
<math display=block>|\langle x, y \rangle| \leq \|x\| \, \|y\|</math>
for every <math>x, y\in V,</math>
with equality if and only if <math>x</math> and <math>y</math> are [[Linearly independent|linearly dependent]].
}}
{{term|[[Parallelogram law]]}}{{defn|
<math display=block>\|x + y\|^2 + \|x - y\|^2 = 2\|x\|^2 + 2\|y\|^2</math>
for every <math>x, y\in V.</math>

The parallelogram law is a necessary and sufficient condition for a norm to be defined by an inner product.
}}
{{term|[[Polarization identity]]}}{{defn|
<math display=block>\|x + y\|^2 = \|x\|^2 + \|y\|^2 + 2\operatorname{Re}\langle x, y \rangle</math>
for every <math>x, y\in V.</math>
The inner product can be retrieved from the norm by the polarization identity, since its imaginary part is the real part of <math>\langle x, iy \rangle.</math>
}}
{{term|[[Ptolemy's inequality]]}}{{defn|
<math display=block>\|x - y\| \, \|z\| ~+~ \|y - z\| \, \|x\| ~\geq~ \|x - z\| \, \|y\|</math>
for every <math>x, y,z\in V.</math>
Ptolemy's inequality is a necessary and sufficient condition for a [[seminorm]] to be the norm defined by an inner product.<ref>{{Cite journal|last=Apostol|first=Tom M.|date=1967|title=Ptolemy's Inequality and the Chordal Metric|url=https://www.tandfonline.com/doi/pdf/10.1080/0025570X.1967.11975804|journal=Mathematics Magazine|volume=40|issue=5|pages=233–235|language=en|doi=10.2307/2688275|jstor=2688275}}</ref>
}}
{{glossary end}}

===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>.
}}
{{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> 
}}
{{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]].
}}
{{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}}

===Real and complex parts of inner products===

Suppose that <math>\langle \cdot, \cdot \rangle</math> is an inner product on <math>V</math> (so it is antilinear in its second argument). The [[polarization identity]] shows that the [[real part]] of the inner product is
<math display=block>\operatorname{Re} \langle x, y \rangle = \frac{1}{4} \left(\|x + y\|^2 - \|x - y\|^2\right).</math>

If <math>V</math> is a real vector space then
<math display=block>\langle x, y \rangle
= \operatorname{Re} \langle x, y \rangle
= \frac{1}{4} \left(\|x + y\|^2 - \|x - y\|^2\right)</math>
and the [[imaginary part]] (also called the {{em|complex part}}) of <math>\langle \cdot, \cdot \rangle</math> is always <math>0.</math>

Assume for the rest of this section that <math>V</math> is a complex vector space.
The [[polarization identity]] for complex vector spaces shows that
<math display="block">\begin{alignat}{4}
\langle x, \ y \rangle
&= \frac{1}{4} \left(\|x + y\|^2 - \|x - y\|^2 + i\|x + iy\|^2 - i\|x - iy\|^2 \right) \\
&= \operatorname{Re} \langle x, y \rangle + i \operatorname{Re} \langle x, i y \rangle. \\
\end{alignat}</math>

The map defined by <math>\langle x \mid y \rangle = \langle y, x \rangle</math> for all <math>x, y \in V</math> satisfies the axioms of the inner product except that it is antilinear in its {{em|first}}, rather than its second, argument. The real part of both <math>\langle x \mid y \rangle</math> and <math>\langle x, y \rangle</math> are equal to <math>\operatorname{Re} \langle x, y \rangle</math> but the inner products differ in their complex part:
<math display="block">\begin{alignat}{4}
\langle x \mid y \rangle
&= \frac{1}{4} \left(\|x + y\|^2 - \|x - y\|^2 - i\|x + iy\|^2 + i\|x - iy\|^2 \right) \\
&= \operatorname{Re} \langle x, y \rangle - i \operatorname{Re} \langle x, i y \rangle. \\
\end{alignat}</math>

The last equality is similar to the formula [[Real and imaginary parts of a linear functional|expressing a linear functional]] in terms of its real part. 

These formulas show that every complex inner product is completely determined by its real part. Moreover, this real part defines an inner product on <math>V,</math> considered as a real vector space. There is thus a one-to-one correspondence between complex inner products on a complex vector space <math>V,</math> and real inner products on <math>V.</math> 

For example, suppose that <math>V = \Complex^n </math> for some integer <math>n > 0.</math> When <math>V</math> is considered as a real vector space in the usual way (meaning that it is identified with the <math>2 n-</math>dimensional real vector space <math>\R^{2n},</math> with each <math>\left(a_1 + i b_1, \ldots, a_n + i b_n\right) \in \Complex^n</math> identified with <math>\left(a_1, b_1, \ldots, a_n, b_n\right) \in \R^{2n}</math>), then the [[dot product]] <math>x \,\cdot\, y = \left(x_1, \ldots, x_{2n}\right) \, \cdot \, \left(y_1, \ldots, y_{2n}\right) := x_1 y_1 + \cdots + x_{2n} y_{2n}</math> defines a real inner product on this space. The unique complex inner product <math>\langle \,\cdot, \cdot\, \rangle</math> on <math>V = \C^n</math> induced by the dot product is the map that sends <math>c = \left(c_1, \ldots, c_n\right), d = \left(d_1, \ldots, d_n\right) \in \Complex^n</math> to <math>\langle c, d \rangle := c_1 \overline{d_1} + \cdots + c_n \overline{d_n}</math> (because the real part of this map <math>\langle \,\cdot, \cdot\, \rangle</math> is equal to the dot product). 

====Real vs. complex inner products====

Let <math>V_{\R}</math> denote <math>V</math> considered as a vector space over the real numbers rather than complex numbers.
The [[real part]] of the complex inner product <math>\langle x, y \rangle</math> is the map <math>\langle x, y \rangle_{\R} = \operatorname{Re} \langle x, y \rangle ~:~ V_{\R} \times V_{\R} \to \R,</math> which necessarily forms a real inner product on the real vector space <math>V_{\R}.</math> Every inner product on a real vector space is a [[Bilinear map|bilinear]] and [[symmetric map]]. 

For example, if <math>V = \Complex</math> with inner product <math>\langle x, y \rangle = x \overline{y},</math> where <math>V</math> is a vector space over the field <math>\Complex,</math> then <math>V_{\R} = \R^2</math> is a vector space over <math>\R</math> and <math>\langle x, y \rangle_{\R}</math> is the [[dot product]] <math>x \cdot y,</math> where <math>x = a + i b \in V = \Complex</math> is identified with the point <math>(a, b) \in V_{\R} = \R^2</math> (and similarly for <math>y</math>); thus the standard inner product <math>\langle x, y \rangle = x \overline{y},</math> on <math>\Complex</math> is an "extension" the dot product . Also, had <math>\langle x, y \rangle</math> been instead defined to be the {{EquationNote|Symmetry|symmetric map}} <math>\langle x, y \rangle = x y</math>  (rather than the usual {{EquationNote|Conjugate symmetry|conjugate symmetric map}} <math>\langle x, y \rangle = x \overline{y}</math>) then its real part <math>\langle x, y \rangle_{\R}</math> would {{em|not}} be the dot product; furthermore, without the complex conjugate, if <math>x \in \C</math> but <math>x \not\in \R</math> then <math>\langle x, x \rangle = x x = x^2 \not\in [0, \infty)</math> so the assignment <math display="inline">x \mapsto \sqrt{\langle x, x \rangle}</math> would not define a norm.

The next examples show that although real and complex inner products have many properties and results in common, they are not entirely interchangeable.
For instance, if <math>\langle x, y \rangle = 0</math> then <math>\langle x, y \rangle_{\R} = 0,</math> but the next example shows that the converse is in general {{em|not}} true.
Given any <math>x \in V,</math> the vector <math>i x</math> (which is the vector <math>x</math> rotated by 90°) belongs to <math>V</math> and so also belongs to <math>V_{\R}</math> (although scalar multiplication of <math>x</math> by <math>i = \sqrt{-1}</math> is not defined in <math>V_{\R},</math> the vector in <math>V</math> denoted by <math>i x</math> is nevertheless still also an element of <math>V_{\R}</math>). For the complex inner product, <math>\langle x, ix \rangle = -i \|x\|^2,</math> whereas for the real inner product the value is always <math>\langle x, ix \rangle_{\R} = 0.</math>

If <math>\langle \,\cdot, \cdot\, \rangle</math> is a complex inner product and <math>A : V \to V</math> is a continuous linear operator that satisfies <math>\langle x, A x \rangle = 0</math> for all <math>x \in V,</math> then <math>A = 0.</math> This statement is no longer true if <math>\langle \,\cdot, \cdot\, \rangle</math> is instead a real inner product, as this next example shows. 
Suppose that <math>V = \Complex</math> has the inner product <math>\langle x, y \rangle := x \overline{y}</math> mentioned above. Then the map <math>A : V \to V</math> defined by <math>A x = ix</math> is a linear map (linear for both <math>V</math> and <math>V_{\R}</math>) that denotes rotation by <math>90^{\circ}</math> in the plane. Because <math>x</math> and <math>A x</math> are perpendicular vectors and <math>\langle x, Ax \rangle_{\R}</math> is just the dot product, <math>\langle x, Ax \rangle_{\R} = 0</math> for all vectors <math>x;</math> nevertheless, this rotation map <math>A</math> is certainly not identically <math>0.</math> In contrast, using the complex inner product gives <math>\langle x, Ax \rangle = -i \|x\|^2,</math> which (as expected) is not identically zero.