Inner product space

In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space^[1]^[2]) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often denoted with angle brackets such as in $\langle a,b\rangle$ . Inner products allow formal definitions of intuitive geometric notions, such as lengths, angles, and orthogonality (zero inner product) of vectors. Inner product spaces generalize Euclidean vector spaces, in which the inner product is the dot product or scalar product of Cartesian coordinates. Inner product spaces of infinite dimension are widely used in functional analysis. Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces. The first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in 1898.^[3]

"Inner product" redirects here. For the inner product of coordinate vectors, see Dot product.

An inner product naturally induces an associated norm, (denoted $|x|$ and $|y|$ in the picture); so, every inner product space is a normed vector space. If this normed space is also complete (that is, a Banach space) then the inner product space is a Hilbert space.^[1] If an inner product space $H$ is not a Hilbert space, it can be extended by completion to a Hilbert space ${\overline {H}}.$ This means that $H$ is a linear subspace of ${\overline {H}},$ the inner product of $H$ is the restriction of that of ${\overline {H}},$ and $H$ is dense in ${\overline {H}}$ for the topology defined by the norm.^[1]^[4]

Conjugate symmetry: $\langle x,y\rangle ={\overline {\langle y,x\rangle }}.$ As ${\textstyle a={\overline {a}}}$ if and only if $a$ is real, conjugate symmetry implies that $\langle x,x\rangle$ is always a real number. If $F$ is $\mathbb {R}$ , conjugate symmetry is just symmetry.

in the first argument:^{[Note 1]} $\langle ax+by,z\rangle =a\langle x,z\rangle +b\langle y,z\rangle .$

Linearity

: if $x$ is not zero, then $\langle x,x\rangle >0$ (conjugate symmetry implies that $\langle x,x\rangle$ is real).

Positive-definiteness

Some examples[edit]

Real and complex numbers[edit]

Among the simplest examples of inner product spaces are $\mathbb {R}$ and $\mathbb {C} .$ The real numbers $\mathbb {R}$ are a vector space over $\mathbb {R}$ that becomes an inner product space with arithmetic multiplication as its inner product: $\langle x,y\rangle :=xy\quad {\text{ for }}x,y\in \mathbb {R} .$

The complex numbers $\mathbb {C}$ are a vector space over $\mathbb {C}$ that becomes an inner product space with the inner product $\langle x,y\rangle :=x{\overline {y}}\quad {\text{ for }}x,y\in \mathbb {C} .$ Unlike with the real numbers, the assignment $(x,y)\mapsto xy$ does not define a complex inner product on $\mathbb {C} .$

Euclidean vector space[edit]

More generally, the real $n$ -space $\mathbb {R} ^{n}$ 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 $x^{\operatorname {T} }$ is the transpose of $x.$

A function $\langle \,\cdot ,\cdot \,\rangle :\mathbb {R} ^{n}\times \mathbb {R} ^{n}\to \mathbb {R}$ is an inner product on $\mathbb {R} ^{n}$ if and only if there exists a symmetric positive-definite matrix $\mathbf {M}$ such that $\langle x,y\rangle =x^{\operatorname {T} }\mathbf {M} y$ for all $x,y\in \mathbb {R} ^{n}.$ If $\mathbf {M}$ is the identity matrix then $\langle x,y\rangle =x^{\operatorname {T} }\mathbf {M} y$ is the dot product. For another example, if $n=2$ and $\mathbf {M} ={\begin{bmatrix}a&b\\b&d\end{bmatrix}}$ is positive-definite (which happens if and only if $\det \mathbf {M} =ad-b^{2}>0$ and one/both diagonal elements are positive) then for any $x:=\left[x_{1},x_{2}\right]^{\operatorname {T} },y:=\left[y_{1},y_{2}\right]^{\operatorname {T} }\in \mathbb {R} ^{2},$ $\langle x,y\rangle :=x^{\operatorname {T} }\mathbf {M} y=\left[x_{1},x_{2}\right]{\begin{bmatrix}a&b\\b&d\end{bmatrix}}{\begin{bmatrix}y_{1}\\y_{2}\end{bmatrix}}=ax_{1}y_{1}+bx_{1}y_{2}+bx_{2}y_{1}+dx_{2}y_{2}.$ As mentioned earlier, every inner product on $\mathbb {R} ^{2}$ is of this form (where $b\in \mathbb {R} ,a>0$ and $d>0$ satisfy $ad>b^{2}$ ).

Complex coordinate space[edit]

The general form of an inner product on $\mathbb {C} ^{n}$ is known as the Hermitian form and is given by $\langle x,y\rangle =y^{\dagger }\mathbf {M} x={\overline {x^{\dagger }\mathbf {M} y}},$ where $M$ is any Hermitian positive-definite matrix and $y^{\dagger }$ 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.

Hilbert space[edit]

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 $\langle f,g\rangle =\int _{a}^{b}f(t){\overline {g(t)}}\,\mathrm {d} t.$ This space is not complete; consider for example, for the interval $[-1, 1]$ the sequence of continuous "step" functions, $\{f_{k}\}_{k},$ defined by: $f_{k}(t)={\begin{cases}0&t\in [-1,0]\\1&t\in \left[{\tfrac {1}{k}},1\right]\\kt&t\in \left(0,{\tfrac {1}{k}}\right)\end{cases}}$

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

Random variables[edit]

For real random variables $X$ and $Y,$ the expected value of their product $\langle X,Y\rangle =\mathbb {E} [XY]$ is an inner product.^[8]^[9]^[10] In this case, $\langle X,X\rangle =0$ if and only if $\mathbb {P} [X=0]=1$ (that is, $X=0$ almost surely), where $\mathbb {P}$ denotes the probability of the event. This definition of expectation as inner product can be extended to random vectors as well.

Complex matrices[edit]

The inner product for complex square matrices of the same size is the Frobenius inner product $\langle A,B\rangle :=\operatorname {tr} \left(AB^{\dagger }\right)$ . 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, $\langle A,B\rangle =\operatorname {tr} \left(AB^{\dagger }\right)={\overline {\operatorname {tr} \left(BA^{\dagger }\right)}}={\overline {\left\langle B,A\right\rangle }}$ Finally, since for $A$ nonzero, $\langle A,A\rangle =\sum _{ij}\left|A_{ij}\right|^{2}>0$ , we get that the Frobenius inner product is positive definite too, and so is an inner product.

Vector spaces with forms[edit]

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

Basic results, terminology, and definitions[edit]

Norm properties [edit]

Every inner product space induces a norm, called its canonical norm, that is defined by $\|x\|={\sqrt {\langle x,x\rangle }}.$ 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:

: $A:V\to W$ is linear and continuous with respect to the metric defined above, or equivalently, $A$ is linear and the set of non-negative reals $\{\|Ax\|:\|x\|\leq 1\},$ where $x$ ranges over the closed unit ball of $V,$ is bounded.

Continuous linear maps

Symmetric linear operators: $A:V\to W$ is linear and $\langle Ax,y\rangle =\langle x,Ay\rangle$ for all $x,y\in V.$

: $A:V\to W$ satisfies $\|Ax\|=\|x\|$ for all $x\in V.$ A linear isometry (resp. an 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 $\langle Ax,Ay\rangle =\langle x,y\rangle$ for all $x,y\in V.$ All isometries are injective. The Mazur–Ulam theorem establishes that every surjective isometry between two 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).

Isometries

Isometrical isomorphisms: $A:V\to W$ is an isometry which is (and hence bijective). Isometrical isomorphisms are also known as unitary operators (compare with unitary matrix).

surjective

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

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.^[13]

Related products[edit]

The term "inner product" is opposed to outer product, which is a slightly more general opposite. Simply, in coordinates, the inner product is the product of a $1\times n$ covector with an $n\times 1$ vector, yielding a $1\times 1$ matrix (a scalar), while the outer product is the product of an $m\times 1$ vector with a $1\times n$ covector, yielding an $m\times n$ 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 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 $W\times V^{*}\to \hom(V,W)$ 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 $V^{*}\times V\to F$ 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 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 outer product (alternatively, wedge product). The inner product is more correctly called a scalar product in this context, as the nondegenerate quadratic form in question need not be positive definite (need not be an inner product).

– Scalar-valued bilinear function

Bilinear form

Biorthogonal system

– In mathematics, vector space of linear forms

Dual space

– subspace of a given real Hilbert space equipped with a new "energetic" inner product

Energetic space

– Generalization of inner products that applies to all normed spaces

L-semi-inner product

– Mathematical metric in normed vector space

Minkowski distance

Orthogonal basis

– Concept in linear algebra

Orthogonal complement

– Specific linear basis (mathematics)

Orthonormal basis

Riemannian manifold

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