Editing Inner product space (section)

==Examples==

===Real and complex numbers===

Among the simplest examples of inner product spaces are <math>\R</math> and <math>\Complex.</math> 
The [[real number]]s <math>\R</math> are a vector space over <math>\R</math> that becomes an inner product space with arithmetic multiplication as its inner product:
<math display=block>\langle x, y \rangle := x y \quad \text{ for } x, y \in \R.</math>

The [[complex number]]s <math>\Complex</math> are a vector space over <math>\Complex</math> that becomes an inner product space with the inner product
<math display=block>\langle x, y \rangle := x \overline{y} \quad \text{ for } x, y \in \Complex.</math> 
Unlike with the real numbers, the assignment <math>(x, y) \mapsto x y</math> does {{em|not}} define a complex inner product on <math>\Complex.</math>

===Euclidean vector space===

More generally, the [[Real coordinate space|real <math>n</math>-space]] <math>\R^n</math> with the [[dot product]] is an inner product space, an example of a [[Euclidean vector space]].
<math display=block>
\left\langle
  \begin{bmatrix} x_1 \\ \vdots \\ x_n \end{bmatrix},
  \begin{bmatrix} y_1 \\ \vdots \\ y_n \end{bmatrix}
\right\rangle 
= x^\textsf{T} y = \sum_{i=1}^n x_i y_i = x_1 y_1 + \cdots + x_n y_n,
</math>
where <math>x^{\operatorname{T}}</math> is the [[transpose]] of <math>x.</math>

A function <math>\langle \,\cdot, \cdot\, \rangle : \R^n \times \R^n \to \R</math> is an inner product on <math>\R^n</math> if and only if there exists a [[Symmetric matrix|symmetric]] [[positive-definite matrix]] <math>\mathbf{M}</math> such that <math>\langle x, y \rangle = x^{\operatorname{T}} \mathbf{M} y</math> for all <math>x, y \in \R^n.</math> If <math>\mathbf{M}</math> is the [[identity matrix]] then <math>\langle x, y \rangle = x^{\operatorname{T}} \mathbf{M} y</math> is the dot product. For another example, if <math>n = 2</math> and <math>\mathbf{M} = \begin{bmatrix} a & b \\ b & d \end{bmatrix}</math> is positive-definite (which happens if and only if <math>\det \mathbf{M} = a d - b^2 > 0</math> and one/both diagonal elements are positive) then for any <math>x := \left[x_1, x_2\right]^{\operatorname{T}}, y := \left[y_1, y_2\right]^{\operatorname{T}} \in \R^2,</math>
<math display=block>\langle x, y \rangle 
:= x^{\operatorname{T}} \mathbf{M} y 
= \left[x_1, x_2\right] \begin{bmatrix} a & b \\ b & d \end{bmatrix} \begin{bmatrix} y_1 \\ y_2 \end{bmatrix} 
= a x_1 y_1 + b x_1 y_2 + b x_2 y_1 + d x_2 y_2.</math> 
As mentioned earlier, every inner product on <math>\R^2</math> is of this form (where <math>b \in \R, a > 0</math> and <math>d > 0</math> satisfy <math>a d > b^2</math>).

===Complex coordinate space===

The general form of an inner product on <math>\Complex^n</math> is known as the [[Hermitian form]] and is given by
<math display=block>\langle x, y \rangle = y^\dagger \mathbf{M} x = \overline{x^\dagger \mathbf{M} y},</math>
where <math>M</math> is any [[Hermitian matrix|Hermitian]] [[positive-definite matrix]] and <math>y^{\dagger}</math> is the [[conjugate transpose]] of <math>y.</math> For the real case, this corresponds to the dot product of the results of directionally-different [[Scaling (geometry)|scaling]] of the two vectors, with positive [[scale factor]]s and orthogonal directions of scaling. It is a [[Weight function|weighted-sum]] version of the dot product with positive weights—up to an orthogonal transformation.

===Hilbert space===

The article on [[Hilbert spaces]] has several examples of inner product spaces, wherein the metric induced by the inner product yields a [[complete metric space]]. An example of an inner product space which induces an incomplete metric is the space <math>C([a, b])</math> of continuous complex valued functions <math>f</math> and <math>g</math> on the interval <math>[a, b].</math> The inner product is
<math display=block>\langle f, g \rangle = \int_a^b f(t) \overline{g(t)} \, \mathrm{d}t.</math>
This space is not complete; consider for example, for the interval {{closed-closed|−1, 1}} the sequence of continuous "step" functions, <math>\{ f_k \}_k,</math> defined by:
<math display=block>f_k(t) = \begin{cases} 0 & t \in [-1, 0] \\ 1 & t \in \left[\tfrac{1}{k}, 1\right] \\ kt & t \in \left(0, \tfrac{1}{k}\right) \end{cases}</math>

This sequence is a [[Cauchy sequence]] for the norm induced by the preceding inner product, which does not converge to a {{em|continuous}} function.

===Random variables===

For real [[random variable]]s <math>X</math> and <math>Y,</math> the [[expected value]] of their product
<math display="block">\langle X, Y \rangle = \mathbb{E}[XY]</math>
is an inner product.<ref>{{cite web|last1=Ouwehand|first1=Peter|title=Spaces of Random Variables|url=http://users.aims.ac.za/~pouw/Lectures/Lecture_Spaces_Random_Variables.pdf|website=AIMS|access-date=2017-09-05|date=November 2010|archive-date=2017-09-05|archive-url=https://web.archive.org/web/20170905225616/http://users.aims.ac.za/~pouw/Lectures/Lecture_Spaces_Random_Variables.pdf|url-status=dead}}</ref><ref>{{cite web|last1=Siegrist|first1=Kyle|title=Vector Spaces of Random Variables|url=http://www.math.uah.edu/stat/expect/Spaces.html|website=Random: Probability, Mathematical Statistics, Stochastic Processes|access-date=2017-09-05|date=1997}}</ref><ref>{{cite thesis|last1=Bigoni|first1=Daniele|title=Uncertainty Quantification with Applications to Engineering Problems|date=2015|type=PhD|publisher=Technical University of Denmark|chapter-url=http://orbit.dtu.dk/files/106969507/phd359_Bigoni_D.pdf|access-date=2017-09-05|chapter=Appendix B: Probability theory and functional spaces}}</ref> In this case, <math>\langle X, X \rangle = 0</math> if and only if <math>\mathbb{P}[X = 0] = 1</math> (that is, <math>X = 0</math> [[almost surely]]), where <math>\mathbb{P}</math> denotes the [[probability]] of the event. This definition of expectation as inner product can be extended to [[random vector]]s as well.

===Complex matrices===

The inner product for complex square matrices of the same size is the [[Frobenius inner product]] <math>\langle A, B \rangle := \operatorname{tr}\left(AB^\dagger\right)</math>. Since trace and transposition are linear and the conjugation is on the second matrix, it is a sesquilinear operator. We further get Hermitian symmetry by, 
<math display=block>\langle A, B \rangle = \operatorname{tr}\left(AB^\dagger\right) = \overline{\operatorname{tr}\left(BA^\dagger\right)} = \overline{\left\langle B,A \right\rangle}</math>
Finally, since for <math>A</math> nonzero, <math>\langle A, A\rangle = \sum_{ij} \left|A_{ij}\right|^2 > 0 </math>, we get that the Frobenius inner product is positive definite too, and so is an inner product.

===Vector spaces with forms===

On an inner product space, or more generally a vector space with a [[nondegenerate form]] (hence an isomorphism <math>V \to V^*</math>), vectors can be sent to covectors (in coordinates, via transpose), so that one can take the inner product and outer product of two vectors—not simply of a vector and a covector.