inner product space#Related products

In this article, {{math|F}} denotes a field that is either the real numbers $\backslash R,$ or the complex numbers $\backslash Complex.$ A scalar is thus an element of {{math|F}}. A bar over an expression representing a scalar denotes the complex conjugate of this scalar. A zero vector is denoted $\backslash mathbf\; 0$ for distinguishing it from the scalar {{math|0}}.

An inner product space is a vector space {{math|V}} over the field {{math|F}} together with an inner product, that is, a map

$$\backslash langle\; \backslash cdot,\; \backslash cdot\; \backslash rangle\; :\; V\; \backslash times\; V\; \backslash to\; F$$

that satisfies the following three properties for all vectors $x,y,z\backslash in\; V$ and all scalars {{nowrap|$a,b\; \backslash in\; F$.{{cite book |title=Functional Analysis |first1=P. K. |last1=Jain |first2=Khalil |last2=Ahmad |chapter-url=https://books.google.com/books?id=yZ68h97pnAkC&pg=PA203 |page=203 |chapter=5.1 Definitions and basic properties of inner product spaces and Hilbert spaces |isbn=81-224-0801-X |year=1995 |edition=2nd |publisher=New Age International}}{{cite book |title=Quantum Mechanics in Hilbert Space |first=Eduard |last=Prugovečki |chapter-url=https://books.google.com/books?id=GxmQxn2PF3IC&pg=PA18 |chapter=Definition 2.1 |pages=18ff |isbn=0-12-566060-X | year = 1981 |publisher=Academic Press |edition = 2nd}}}}

• Conjugate symmetry: $$\backslash langle\; x,\; y\; \backslash rangle\; =\; \backslash overline\{\backslash langle\; y,\; x\; \backslash rangle\}.$$ As $a\; =\; \backslash overline\{a\}$ if and only if $a$ is real, conjugate symmetry implies that $\backslash langle\; x,\; x\; \backslash rangle$ is always a real number. If {{math|F}} is $\backslash R$, conjugate symmetry is just symmetry.
• Linearity in the first argument:By combining the linear in the first argument property with the conjugate symmetry property you get conjugate-linear in the second argument: $\backslash langle\; x,by\; \backslash rangle\; =\; \backslash langle\; x,y\; \backslash rangle\; \backslash overline\{b\}$. This is how the inner product was originally defined and is used in most mathematical contexts. A different convention has been adopted in theoretical physics and quantum mechanics, originating in the bra-ket notation of Paul Dirac, where the inner product is taken to be linear in the second argument and conjugate-linear in the first argument; this convention is used in many other domains such as engineering and computer science. $$$$

\langle ax+by, z \rangle = a \langle x, z \rangle + b \langle y, z \rangle.

• Positive-definiteness: if $x$ is not zero, then $$$$

\langle x, x \rangle > 0

(conjugate symmetry implies that $\backslash langle\; x,\; x\; \backslash rangle$ is real).

If the positive-definiteness condition is replaced by merely requiring that $\backslash langle\; x,\; x\; \backslash rangle\; \backslash geq\; 0$ for all $x$, then one obtains the definition of positive semi-definite Hermitian form. A positive semi-definite Hermitian form $\backslash langle\; \backslash cdot,\; \backslash cdot\; \backslash rangle$ is an inner product if and only if for all $x$, if $\backslash langle\; x,\; x\; \backslash rangle\; =\; 0$ then $x\; =\; \backslash mathbf\; 0$.{{sfn|Schaefer|Wolff|1999|p=44}}

In the following properties, which result almost immediately from the definition of an inner product, {{math|x, y}} and {{mvar|z}} are arbitrary vectors, and {{mvar|a}} and {{mvar|b}} are arbitrary scalars.

• $\backslash langle\; \backslash mathbf\{0\},\; x\; \backslash rangle=\backslash langle\; x,\backslash mathbf\{0\}\backslash rangle=0.$
• $\backslash langle\; x,\; x\; \backslash rangle$ is real and nonnegative.
• $\backslash langle\; x,\; x\; \backslash rangle\; =\; 0$ if and only if $x=\backslash mathbf\{0\}.$
• $\backslash langle\; x,\; ay+bz\; \backslash rangle=\; \backslash overline\; a\; \backslash langle\; x,\; y\; \backslash rangle\; +\; \backslash overline\; b\; \backslash langle\; x,\; z\; \backslash rangle.$
  This implies that an inner product is a sesquilinear form.
• $\backslash langle\; x\; +\; y,\; x\; +\; y\; \backslash rangle\; =\; \backslash langle\; x,\; x\; \backslash rangle\; +\; 2\backslash operatorname\{Re\}(\backslash langle\; x,\; y\; \backslash rangle)\; +\; \backslash langle\; y,\; y\; \backslash rangle,$ where $\backslash operatorname\{Re\}$
  denotes the real part of its argument.

Over $\backslash R$, conjugate-symmetry reduces to symmetry, and sesquilinearity reduces to bilinearity. Hence an inner product on a real vector space is a positive-definite symmetric bilinear form. The binomial expansion of a square becomes

$$\backslash langle\; x\; +\; y,\; x\; +\; y\; \backslash rangle\; =\; \backslash langle\; x,\; x\; \backslash rangle\; +\; 2\backslash langle\; x,\; y\; \backslash rangle\; +\; \backslash langle\; y,\; y\; \backslash rangle\; .$$

Several notations are used for inner products, including

$\backslash langle\; \backslash cdot,\; \backslash cdot\; \backslash rangle$,

$\backslash left\; (\; \backslash cdot,\; \backslash cdot\; \backslash right\; )$,

$\backslash langle\; \backslash cdot\; |\; \backslash cdot\; \backslash rangle$ and

$\backslash left\; (\; \backslash cdot\; |\; \backslash cdot\; \backslash right\; )$, as well as the usual dot product.

Some authors, especially in physics and matrix algebra, prefer to define inner products and sesquilinear forms with linearity in the second argument rather than the first. Then the first argument becomes conjugate linear, rather than the second. Bra-ket notation in quantum mechanics also uses slightly different notation, i.e. $\backslash langle\; \backslash cdot\; |\; \backslash cdot\; \backslash rangle$, where $\backslash langle\; x\; |\; y\; \backslash rangle\; :=\; \backslash left\; (\; y,\; x\; \backslash right\; )$.

Among the simplest examples of inner product spaces are $\backslash R$ and $\backslash Complex.$

The real numbers $\backslash R$ are a vector space over $\backslash R$ that becomes an inner product space with arithmetic multiplication as its inner product:

$$\backslash langle\; x,\; y\; \backslash rangle\; :=\; x\; y\; \backslash quad\; \backslash text\{\; for\; \}\; x,\; y\; \backslash in\; \backslash R.$$

The complex numbers $\backslash Complex$ are a vector space over $\backslash Complex$ that becomes an inner product space with the inner product

$$\backslash langle\; x,\; y\; \backslash rangle\; :=\; x\; \backslash overline\{y\}\; \backslash quad\; \backslash text\{\; for\; \}\; x,\; y\; \backslash in\; \backslash Complex.$$

Unlike with the real numbers, the assignment $(x,\; y)\; \backslash mapsto\; x\; y$ does {{em|not}} define a complex inner product on $\backslash Complex.$

More generally, the real $n$-space $\backslash R^n$ with the dot product is an inner product space, an example of a Euclidean vector space.

\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,

where $x^\{\backslash operatorname\{T\}\}$ is the transpose of $x.$

A function $\backslash langle\; \backslash ,\backslash cdot,\; \backslash cdot\backslash ,\; \backslash rangle\; :\; \backslash R^n\; \backslash times\; \backslash R^n\; \backslash to\; \backslash R$ is an inner product on $\backslash R^n$ if and only if there exists a symmetric positive-definite matrix $\backslash mathbf\{M\}$ such that $\backslash langle\; x,\; y\; \backslash rangle\; =\; x^\{\backslash operatorname\{T\}\}\; \backslash mathbf\{M\}\; y$ for all $x,\; y\; \backslash in\; \backslash R^n.$ If $\backslash mathbf\{M\}$ is the identity matrix then $\backslash langle\; x,\; y\; \backslash rangle\; =\; x^\{\backslash operatorname\{T\}\}\; \backslash mathbf\{M\}\; y$ is the dot product. For another example, if $n\; =\; 2$ and is positive-definite (which happens if and only if $\backslash det\; \backslash mathbf\{M\}\; =\; a\; d\; -\; b^2\; >\; 0$ and one/both diagonal elements are positive) then for any $x\; :=\; \backslash left[x\_1,\; x\_2\backslash right]^\{\backslash operatorname\{T\}\},\; y\; :=\; \backslash left[y\_1,\; y\_2\backslash right]^\{\backslash operatorname\{T\}\}\; \backslash in\; \backslash R^2,$

$$\backslash langle\; x,\; y\; \backslash rangle$$

:= 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.

As mentioned earlier, every inner product on $\backslash R^2$ is of this form (where $b\; \backslash in\; \backslash R,\; a\; >\; 0$ and $d\; >\; 0$ satisfy $a\; d\; >\; b^2$).

The general form of an inner product on $\backslash Complex^n$ is known as the Hermitian form and is given by

$$\backslash langle\; x,\; y\; \backslash rangle\; =\; y^\backslash dagger\; \backslash mathbf\{M\}\; x\; =\; \backslash overline\{x^\backslash dagger\; \backslash mathbf\{M\}\; y\},$$

where $M$ is any Hermitian positive-definite matrix and $y^\{\backslash dagger\}$ is the conjugate transpose of $y.$ For the real case, this corresponds to the dot product of the results of directionally-different scaling of the two vectors, with positive scale factors and orthogonal directions of scaling. It is a weighted-sum version of the dot product with positive weights—up to an orthogonal transformation.

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 $C([a,\; b])$ of continuous complex valued functions $f$ and $g$ on the interval $[a,\; b].$ The inner product is

$$\backslash langle\; f,\; g\; \backslash rangle\; =\; \backslash int\_a^b\; f(t)\; \backslash overline\{g(t)\}\; \backslash ,\; \backslash mathrm\{d\}t.$$

This space is not complete; consider for example, for the interval {{closed-closed|−1, 1}} the sequence of continuous "step" functions, $\backslash \{\; f\_k\; \backslash \}\_k,$ defined by:

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

For real random variables $X$ and $Y,$ the expected value of their product

$$\backslash langle\; X,\; Y\; \backslash rangle\; =\; \backslash mathbb\{E\}[XY]$$

is an inner product.{{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}}{{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}}{{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}} In this case, $\backslash langle\; X,\; X\; \backslash rangle\; =\; 0$ if and only if $\backslash mathbb\{P\}[X\; =\; 0]\; =\; 1$ (that is, $X\; =\; 0$ almost surely), where $\backslash mathbb\{P\}$ denotes the probability of the event. This definition of expectation as inner product can be extended to random vectors as well.

The inner product for complex square matrices of the same size is the Frobenius inner product $\backslash langle\; A,\; B\; \backslash rangle\; :=\; \backslash operatorname\{tr\}\backslash left(AB^\backslash dagger\backslash right)$. 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,

$$\backslash langle\; A,\; B\; \backslash rangle\; =\; \backslash operatorname\{tr\}\backslash left(AB^\backslash dagger\backslash right)\; =\; \backslash overline\{\backslash operatorname\{tr\}\backslash left(BA^\backslash dagger\backslash right)\}\; =\; \backslash overline\{\backslash left\backslash langle\; B,A\; \backslash right\backslash rangle\}$$

Finally, since for $A$ nonzero, $\backslash langle\; A,\; A\backslash rangle\; =\; \backslash sum\_\{ij\}\; \backslash left|A\_\{ij\}\backslash right|^2\; >\; 0$, we get that the Frobenius inner product is positive definite too, and so is an inner product.

On an inner product space, or more generally a vector space with a nondegenerate form (hence an isomorphism $V\; \backslash to\; V^*$), 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.

===Norm properties {{anchor|Norm}}===

Every inner product space induces a norm, called its {{em|{{visible anchor|canonical norm}}}}, that is defined by

$$\backslash |x\backslash |\; =\; \backslash sqrt\{\backslash langle\; x,\; x\; \backslash rangle\}.$$

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, one has the following properties:

{{glossary}}

{{term|Absolute homogeneity}}{{defn|

$$\backslash |ax\backslash |\; =\; |a|\; \backslash ,\; \backslash |x\backslash |$$

for every $x\; \backslash in\; V$ and $a\; \backslash in\; F$

(this results from $\backslash langle\; ax,\; ax\; \backslash rangle\; =\; a\backslash overline\; a\; \backslash langle\; x,\; x\; \backslash rangle$).

}}

{{term|Triangle inequality}}{{defn|

$$\backslash |x\; +\; y\backslash |\; \backslash leq\; \backslash |x\backslash |\; +\; \backslash |y\backslash |$$

for $x,\; y\backslash in\; V.$

These two properties show that one has indeed a norm.}}

{{term|Cauchy–Schwarz inequality}}{{defn|

$$|\backslash langle\; x,\; y\; \backslash rangle|\; \backslash leq\; \backslash |x\backslash |\; \backslash ,\; \backslash |y\backslash |$$

for every $x,\; y\backslash in\; V,$

with equality if and only if $x$ and $y$ are linearly dependent.

}}

{{term|Parallelogram law}}{{defn|

$$\backslash |x\; +\; y\backslash |^2\; +\; \backslash |x\; -\; y\backslash |^2\; =\; 2\backslash |x\backslash |^2\; +\; 2\backslash |y\backslash |^2$$

for every $x,\; y\backslash in\; V.$

The parallelogram law is a necessary and sufficient condition for a norm to be defined by an inner product.

}}

{{term|Polarization identity}}{{defn|

$$\backslash |x\; +\; y\backslash |^2\; =\; \backslash |x\backslash |^2\; +\; \backslash |y\backslash |^2\; +\; 2\backslash operatorname\{Re\}\backslash langle\; x,\; y\; \backslash rangle$$

for every $x,\; y\backslash in\; V.$

The inner product can be retrieved from the norm by the polarization identity, since its imaginary part is the real part of $\backslash langle\; x,\; iy\; \backslash rangle.$

}}

{{term|Ptolemy's inequality}}{{defn|

$$\backslash |x\; -\; y\backslash |\; \backslash ,\; \backslash |z\backslash |\; ~+~\; \backslash |y\; -\; z\backslash |\; \backslash ,\; \backslash |x\backslash |\; ~\backslash geq~\; \backslash |x\; -\; z\backslash |\; \backslash ,\; \backslash |y\backslash |$$

for every $x,\; y,z\backslash in\; V.$

Ptolemy's inequality is a necessary and sufficient condition for a seminorm to be the norm defined by an inner product.{{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}}

}}

{{glossary end}}

{{glossary}}

{{term|Orthogonality}}{{defn|

Two vectors $x$ and $y$ are said to be {{em|{{visible anchor|orthogonal|Orthogonal vectors}}}}, often written $x\; \backslash perp\; y,$ if their inner product is zero, that is, if $\backslash langle\; x,\; y\; \backslash rangle\; =\; 0.$

This happens if and only if $\backslash |x\backslash |\; \backslash leq\; \backslash |x\; +\; s\; y\backslash |$ for all scalars $s,${{sfn|Rudin|1991|pp=306-312}} and if and only if the real-valued function $f(s)\; :=\; \backslash |x\; +\; s\; y\backslash |^2\; -\; \backslash |x\backslash |^2$ is non-negative. (This is a consequence of the fact that, if $y\; \backslash neq\; 0$ then the scalar $s\_0\; =\; -\; \backslash tfrac\{\backslash overline\{\backslash langle\; x,\; y\; \backslash rangle\}\}\{\backslash |y\backslash |^2\}$ minimizes $f$ with value $f\backslash left(s\_0\backslash right)\; =\; -\; \backslash tfrac$

$$\backslash angle(x,\; y)\; =\; \backslash arccos\; \backslash frac\{\backslash langle\; x,\; y\; \backslash rangle\}\{\backslash |x\backslash |\; \backslash ,\; \backslash |y\backslash |\},$$

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 $\backslash langle\; x,\; y\; \backslash rangle$ is negative, the angle $\backslash angle(x,\; y)$ 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}}

Suppose that $\backslash langle\; \backslash cdot,\; \backslash cdot\; \backslash rangle$ is an inner product on $V$ (so it is antilinear in its second argument). The polarization identity shows that the real part of the inner product is

$$\backslash operatorname\{Re\}\; \backslash langle\; x,\; y\; \backslash rangle\; =\; \backslash frac\{1\}\{4\}\; \backslash left(\backslash |x\; +\; y\backslash |^2\; -\; \backslash |x\; -\; y\backslash |^2\backslash right).$$

If $V$ is a real vector space then

$$\backslash langle\; x,\; y\; \backslash rangle$$

= \operatorname{Re} \langle x, y \rangle

= \frac{1}{4} \left(\|x + y\|^2 - \|x - y\|^2\right)

and the imaginary part (also called the {{em|complex part}}) of $\backslash langle\; \backslash cdot,\; \backslash cdot\; \backslash rangle$ is always $0.$

Assume for the rest of this section that $V$ is a complex vector space.

The polarization identity for complex vector spaces shows that

$$\backslash begin\{alignat\}\{4\}$$

\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}

The map defined by $\backslash langle\; x\; \backslash mid\; y\; \backslash rangle\; =\; \backslash langle\; y,\; x\; \backslash rangle$ for all $x,\; y\; \backslash in\; V$ 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 $\backslash langle\; x\; \backslash mid\; y\; \backslash rangle$ and $\backslash langle\; x,\; y\; \backslash rangle$ are equal to $\backslash operatorname\{Re\}\; \backslash langle\; x,\; y\; \backslash rangle$ but the inner products differ in their complex part:

$$\backslash begin\{alignat\}\{4\}$$

\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}

The last equality is similar to the formula 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 $V,$ considered as a real vector space. There is thus a one-to-one correspondence between complex inner products on a complex vector space $V,$ and real inner products on $V.$

For example, suppose that $V\; =\; \backslash Complex^n$ for some integer $n\; >\; 0.$ When $V$ is considered as a real vector space in the usual way (meaning that it is identified with the $2\; n-$dimensional real vector space $\backslash R^\{2n\},$ with each $\backslash left(a\_1\; +\; i\; b\_1,\; \backslash ldots,\; a\_n\; +\; i\; b\_n\backslash right)\; \backslash in\; \backslash Complex^n$ identified with $\backslash left(a\_1,\; b\_1,\; \backslash ldots,\; a\_n,\; b\_n\backslash right)\; \backslash in\; \backslash R^\{2n\}$), then the dot product $x\; \backslash ,\backslash cdot\backslash ,\; y\; =\; \backslash left(x\_1,\; \backslash ldots,\; x\_\{2n\}\backslash right)\; \backslash ,\; \backslash cdot\; \backslash ,\; \backslash left(y\_1,\; \backslash ldots,\; y\_\{2n\}\backslash right)\; :=\; x\_1\; y\_1\; +\; \backslash cdots\; +\; x\_\{2n\}\; y\_\{2n\}$ defines a real inner product on this space. The unique complex inner product $\backslash langle\; \backslash ,\backslash cdot,\; \backslash cdot\backslash ,\; \backslash rangle$ on $V\; =\; \backslash C^n$ induced by the dot product is the map that sends $c\; =\; \backslash left(c\_1,\; \backslash ldots,\; c\_n\backslash right),\; d\; =\; \backslash left(d\_1,\; \backslash ldots,\; d\_n\backslash right)\; \backslash in\; \backslash Complex^n$ to $\backslash langle\; c,\; d\; \backslash rangle\; :=\; c\_1\; \backslash overline\{d\_1\}\; +\; \backslash cdots\; +\; c\_n\; \backslash overline\{d\_n\}$ (because the real part of this map $\backslash langle\; \backslash ,\backslash cdot,\; \backslash cdot\backslash ,\; \backslash rangle$ is equal to the dot product).

Let $V\_\{\backslash R\}$ denote $V$ considered as a vector space over the real numbers rather than complex numbers.

The real part of the complex inner product $\backslash langle\; x,\; y\; \backslash rangle$ is the map $\backslash langle\; x,\; y\; \backslash rangle\_\{\backslash R\}\; =\; \backslash operatorname\{Re\}\; \backslash langle\; x,\; y\; \backslash rangle\; ~:~\; V\_\{\backslash R\}\; \backslash times\; V\_\{\backslash R\}\; \backslash to\; \backslash R,$ which necessarily forms a real inner product on the real vector space $V\_\{\backslash R\}.$ Every inner product on a real vector space is a bilinear and symmetric map.

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

Given any $x\; \backslash in\; V,$ the vector $i\; x$ (which is the vector $x$ rotated by 90°) belongs to $V$ and so also belongs to $V\_\{\backslash R\}$ (although scalar multiplication of $x$ by $i\; =\; \backslash sqrt\{-1\}$ is not defined in $V\_\{\backslash R\},$ the vector in $V$ denoted by $i\; x$ is nevertheless still also an element of $V\_\{\backslash R\}$). For the complex inner product, $\backslash langle\; x,\; ix\; \backslash rangle\; =\; -i\; \backslash |x\backslash |^2,$ whereas for the real inner product the value is always $\backslash langle\; x,\; ix\; \backslash rangle\_\{\backslash R\}\; =\; 0.$

If $\backslash langle\; \backslash ,\backslash cdot,\; \backslash cdot\backslash ,\; \backslash rangle$ is a complex inner product and $A\; :\; V\; \backslash to\; V$ is a continuous linear operator that satisfies $\backslash langle\; x,\; A\; x\; \backslash rangle\; =\; 0$ for all $x\; \backslash in\; V,$ then $A\; =\; 0.$ This statement is no longer true if $\backslash langle\; \backslash ,\backslash cdot,\; \backslash cdot\backslash ,\; \backslash rangle$ is instead a real inner product, as this next example shows.

Suppose that $V\; =\; \backslash Complex$ has the inner product $\backslash langle\; x,\; y\; \backslash rangle\; :=\; x\; \backslash overline\{y\}$ mentioned above. Then the map $A\; :\; V\; \backslash to\; V$ defined by $A\; x\; =\; ix$ is a linear map (linear for both $V$ and $V\_\{\backslash R\}$) that denotes rotation by $90^\{\backslash circ\}$ in the plane. Because $x$ and $A\; x$ are perpendicular vectors and $\backslash langle\; x,\; Ax\; \backslash rangle\_\{\backslash R\}$ is just the dot product, $\backslash langle\; x,\; Ax\; \backslash rangle\_\{\backslash R\}\; =\; 0$ for all vectors $x;$ nevertheless, this rotation map $A$ is certainly not identically $0.$ In contrast, using the complex inner product gives $\backslash langle\; x,\; Ax\; \backslash rangle\; =\; -i\; \backslash |x\backslash |^2,$ which (as expected) is not identically zero.

{{See also|Orthogonal basis|Orthonormal basis}}

Let $V$ be a finite dimensional inner product space of dimension $n.$ Recall that every basis of $V$ consists of exactly $n$ linearly independent vectors. Using the Gram–Schmidt process we may start with an arbitrary basis and transform it into an orthonormal basis. That is, into a basis in which all the elements are orthogonal and have unit norm. In symbols, a basis $\backslash \{e\_1,\; \backslash ldots,\; e\_n\backslash \}$ is orthonormal if $\backslash langle\; e\_i,\; e\_j\; \backslash rangle\; =\; 0$ for every $i\; \backslash neq\; j$ and $\backslash langle\; e\_i,\; e\_i\; \backslash rangle\; =\; \backslash |e\_a\backslash |^2\; =\; 1$ for each index $i.$

This definition of orthonormal basis generalizes to the case of infinite-dimensional inner product spaces in the following way. Let $V$ be any inner product space. Then a collection

$$E\; =\; \backslash left\backslash \{\; e\_a\; \backslash right\backslash \}\_\{a\; \backslash in\; A\}$$

is a {{em|basis}} for $V$ if the subspace of $V$ generated by finite linear combinations of elements of $E$ is dense in $V$ (in the norm induced by the inner product). Say that $E$ is an {{em|orthonormal basis}} for $V$ if it is a basis and

$$\backslash left\backslash langle\; e\_\{a\},\; e\_\{b\}\; \backslash right\backslash rangle\; =\; 0$$

if $a\; \backslash neq\; b$ and $\backslash langle\; e\_a,\; e\_a\; \backslash rangle\; =\; \backslash |e\_a\backslash |^2\; =\; 1$ for all $a,\; b\; \backslash in\; A.$

Using an infinite-dimensional analog of the Gram-Schmidt process one may show:

Theorem. Any separable inner product space has an orthonormal basis.

Using the Hausdorff maximal principle and the fact that in a complete inner product space orthogonal projection onto linear subspaces is well-defined, one may also show that

Theorem. Any complete inner product space has an orthonormal basis.

The two previous theorems raise the question of whether all inner product spaces have an orthonormal basis. The answer, it turns out is negative. This is a non-trivial result, and is proved below. The following proof is taken from Halmos's A Hilbert Space Problem Book (see the references).{{citation needed|date=October 2017}}

:

Parseval's identity leads immediately to the following theorem:

Theorem. Let $V$ be a separable inner product space and $\backslash left\backslash \{e\_k\backslash right\backslash \}\_k$ an orthonormal basis of $V.$ Then the map

$$x\; \backslash mapsto\; \backslash bigl\backslash \{\backslash langle\; e\_k,\; x\; \backslash rangle\backslash bigr\backslash \}\_\{k\; \backslash in\; \backslash N\}$$

is an isometric linear map $V\; \backslash rightarrow\; \backslash ell^2$ with a dense image.

This theorem can be regarded as an abstract form of Fourier series, in which an arbitrary orthonormal basis plays the role of the sequence of trigonometric polynomials. Note that the underlying index set can be taken to be any countable set (and in fact any set whatsoever, provided $\backslash ell^2$ is defined appropriately, as is explained in the article Hilbert space). In particular, we obtain the following result in the theory of Fourier series:

Theorem. Let $V$ be the inner product space $C[-\backslash pi,\; \backslash pi].$ Then the sequence (indexed on set of all integers) of continuous functions

$$e\_k(t)\; =\; \backslash frac\{e^\{i\; k\; t\}\}\{\backslash sqrt\{2\; \backslash pi\}\}$$

is an orthonormal basis of the space $C[-\backslash pi,\; \backslash pi]$ with the $L^2$ inner product. The mapping

$$f\; \backslash mapsto\; \backslash frac\{1\}\{\backslash sqrt\{2\; \backslash pi\}\}\; \backslash left\backslash \{\backslash int\_\{-\backslash pi\}^\backslash pi\; f(t)\; e^\{-i\; k\; t\}\; \backslash ,\; \backslash mathrm\{d\}t\; \backslash right\backslash \}\_\{k\; \backslash in\; \backslash Z\}$$

is an isometric linear map with dense image.

Orthogonality of the sequence $\backslash \{\; e\_k\; \backslash \}\_k$ follows immediately from the fact that if $k\; \backslash neq\; j,$ then

$$\backslash int\_\{-\backslash pi\}^\backslash pi\; e^\{-i\; (j\; -\; k)\; t\}\; \backslash ,\; \backslash mathrm\{d\}t\; =\; 0.$$

Normality of the sequence is by design, that is, the coefficients are so chosen so that the norm comes out to 1. Finally the fact that the sequence has a dense algebraic span, in the {{em|inner product norm}}, follows from the fact that the sequence has a dense algebraic span, this time in the space of continuous periodic functions on $[-\backslash pi,\; \backslash pi]$ with the uniform norm. This is the content of the Weierstrass theorem on the uniform density of trigonometric polynomials.

{{Main|Operator theory}}

Several types of linear maps $A\; :\; V\; \backslash to\; W$ between inner product spaces $V$ and $W$ are of relevance:

• {{em|Continuous linear maps}}: $A\; :\; V\; \backslash to\; W$ is linear and continuous with respect to the metric defined above, or equivalently, $A$ is linear and the set of non-negative reals $\backslash \{\; \backslash |Ax\backslash |\; :\; \backslash |x\backslash |\; \backslash leq\; 1\backslash \},$ where $x$ ranges over the closed unit ball of $V,$ is bounded.
• {{em|Symmetric linear operators}}: $A\; :\; V\; \backslash to\; W$ is linear and $\backslash langle\; Ax,\; y\; \backslash rangle\; =\; \backslash langle\; x,\; Ay\; \backslash rangle$ for all $x,\; y\; \backslash in\; V.$
• {{em|Isometries}}: $A\; :\; V\; \backslash to\; W$ satisfies $\backslash |A\; x\backslash |\; =\; \backslash |x\backslash |$ for all $x\; \backslash in\; V.$ A {{em|linear isometry}} (resp. an {{em|antilinear isometry}}) is an isometry that is also a linear map (resp. an antilinear map). For inner product spaces, the polarization identity can be used to show that $A$ is an isometry if and only if $\backslash langle\; Ax,\; Ay\; \backslash rangle\; =\; \backslash langle\; x,\; y\; \backslash rangle$ for all $x,\; y\; \backslash in\; V.$ All isometries are injective. The Mazur–Ulam theorem establishes that every surjective isometry between two {{em|real}} normed spaces is an affine transformation. Consequently, an isometry $A$ between real inner product spaces is a linear map if and only if $A(0)\; =\; 0.$ Isometries are morphisms between inner product spaces, and morphisms of real inner product spaces are orthogonal transformations (compare with orthogonal matrix).
• {{em|Isometrical isomorphisms}}: $A\; :\; V\; \backslash to\; W$ is an isometry which is surjective (and hence bijective). Isometrical isomorphisms are also known as unitary operators (compare with unitary matrix).

From the point of view of inner product space theory, there is no need to distinguish between two spaces which are isometrically isomorphic. The spectral theorem provides a canonical form for symmetric, unitary and more generally normal operators on finite dimensional inner product spaces. A generalization of the spectral theorem holds for continuous normal operators in Hilbert spaces.{{harvnb|Rudin|1991}}

Any of the axioms of an inner product may be weakened, yielding generalized notions. The generalizations that are closest to inner products occur where bilinearity and conjugate symmetry are retained, but positive-definiteness is weakened.

{{Main|Krein space}}

If $V$ is a vector space and $\backslash langle\; \backslash ,\backslash cdot\backslash ,,\; \backslash ,\backslash cdot\backslash ,\; \backslash rangle$ a semi-definite sesquilinear form, then the function:

$$\backslash |x\backslash |\; =\; \backslash sqrt\{\backslash langle\; x,\; x\backslash rangle\}$$

makes sense and satisfies all the properties of norm except that $\backslash |x\backslash |\; =\; 0$ does not imply $x\; =\; 0$ (such a functional is then called a semi-norm). We can produce an inner product space by considering the quotient $W\; =\; V\; /\; \backslash \{x\; :\; \backslash |x\backslash |\; =\; 0\backslash \}.$ The sesquilinear form $\backslash langle\; \backslash ,\backslash cdot\backslash ,,\; \backslash ,\backslash cdot\backslash ,\; \backslash rangle$ factors through $W.$

This construction is used in numerous contexts. The Gelfand–Naimark–Segal construction is a particularly important example of the use of this technique. Another example is the representation of semi-definite kernels on arbitrary sets.

{{Main|Pseudo-Euclidean space}}

Alternatively, one may require that the pairing be a nondegenerate form, meaning that for all non-zero $x\; \backslash neq\; 0$ there exists some $y$ such that $\backslash langle\; x,\; y\; \backslash rangle\; \backslash neq\; 0,$ though $y$ need not equal $x$; in other words, the induced map to the dual space $V\; \backslash to\; V^*$ is injective. This generalization is important in differential geometry: a manifold whose tangent spaces have an inner product is a Riemannian manifold, while if this is related to nondegenerate conjugate symmetric form the manifold is a pseudo-Riemannian manifold. By Sylvester's law of inertia, just as every inner product is similar to the dot product with positive weights on a set of vectors, every nondegenerate conjugate symmetric form is similar to the dot product with {{em|nonzero}} weights on a set of vectors, and the number of positive and negative weights are called respectively the positive index and negative index. Product of vectors in Minkowski space is an example of indefinite inner product, although, technically speaking, it is not an inner product according to the standard definition above. Minkowski space has four dimensions and indices 3 and 1 (assignment of "+" and "−" to them differs depending on conventions).

Purely algebraic statements (ones that do not use positivity) usually only rely on the nondegeneracy (the injective homomorphism $V\; \backslash to\; V^*$) and thus hold more generally.

The term "inner product" is opposed to outer product (tensor product), which is a slightly more general opposite. Simply, in coordinates, the inner product is the product of a $1\; \backslash times\; n$ {{em|covector}} with an $n\; \backslash times\; 1$ vector, yielding a $1\; \backslash times\; 1$ matrix (a scalar), while the outer product is the product of an $m\; \backslash times\; 1$ vector with a $1\; \backslash times\; n$ covector, yielding an $m\; \backslash times\; n$ matrix. The outer product is defined for different dimensions, while the inner product requires the same dimension. If the dimensions are the same, then the inner product is the {{em|trace}} of the outer product (trace only being properly defined for square matrices). In an informal summary: "inner is horizontal times vertical and shrinks down, outer is vertical times horizontal and expands out".

More abstractly, the outer product is the bilinear map $W\; \backslash times\; V^*\; \backslash to\; \backslash hom(V,\; W)$ sending a vector and a covector to a rank 1 linear transformation (simple tensor of type (1, 1)), while the inner product is the bilinear evaluation map $V^*\; \backslash times\; V\; \backslash to\; F$ given by evaluating a covector on a vector; the order of the domain vector spaces here reflects the covector/vector distinction.

The inner product and outer product should not be confused with the interior product and exterior product, which are instead operations on vector fields and differential forms, or more generally on the exterior algebra.

As a further complication, in geometric algebra the inner product and the {{em|exterior}} (Grassmann) product are combined in the geometric product (the Clifford product in a Clifford algebra) – the inner product sends two vectors (1-vectors) to a scalar (a 0-vector), while the exterior product sends two vectors to a bivector (2-vector) – and in this context the exterior product is usually called the {{em|outer product}} (alternatively, {{em|wedge product}}). The inner product is more correctly called a {{em|scalar}} product in this context, as the nondegenerate quadratic form in question need not be positive definite (need not be an inner product).

• {{annotated link|Bilinear form}}
• {{annotated link|Biorthogonal system}}
• {{annotated link|Dual space}}
• {{annotated link|Energetic space}}
• {{annotated link|L-semi-inner product}}
• {{annotated link|Minkowski distance}}
• {{annotated link|Orthogonal basis}}
• {{annotated link|Orthogonal complement}}
• {{annotated link|Orthonormal basis}}
• Riemannian manifold

{{reflist|group="Note"|refs=

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{{reflist}}

• {{Cite book|last1=Axler|first1=Sheldon|title=Linear Algebra Done Right|publisher=Springer-Verlag|location=Berlin, New York|edition=2nd|isbn=978-0-387-98258-8|year=1997}}
• {{cite book|first=Jean|last=Dieudonné|author-link=Jean Dieudonné|title=Treatise on Analysis, Vol. I [Foundations of Modern Analysis]|publisher=Academic Press|year=1969|isbn=978-1-4067-2791-3|edition=2nd}}
• {{Cite book|last1=Emch|first1=Gerard G.|title=Algebraic Methods in Statistical Mechanics and Quantum Field Theory|publisher=Wiley-Interscience|isbn=978-0-471-23900-0|year=1972}}
• {{Halmos A Hilbert Space Problem Book 1982}}
• {{Lax Functional Analysis}}
• {{Rudin Walter Functional Analysis|edition=2}}
• {{Schaefer Wolff Topological Vector Spaces|edition=2}}
• {{Schechter Handbook of Analysis and Its Foundations}}
• {{Swartz An Introduction to Functional Analysis}}
• {{Trèves François Topological vector spaces, distributions and kernels}}
• {{Cite book|last1=Young|first1=Nicholas|title=An Introduction to Hilbert Space|publisher=Cambridge University Press|isbn=978-0-521-33717-5|year=1988}}
• Zamani, A.; Moslehian, M.S.; & Frank, M. (2015) "Angle Preserving Mappings", Journal of Analysis and Applications 34: 485 to 500 {{doi|10.4171/ZAA/1551}}

{{linear algebra}}

{{Functional analysis}}

{{HilbertSpace}}

Category:Normed spaces

Category:Bilinear forms