Inner product space

In mathematics, an inner product space is a vector space with additional structure, an inner product, scalar product or dot product, which allows us to introduce geometrical notions such as angles and lengths of vectors. Inner product spaces are generalizations of Euclidean space (with the dot product as the inner product) and are studied in functional analysis. An inner product space is also called a pre-Hilbert space, since its completion with respect to the metric induced by its inner product is a Hilbert space.

Definitions

In the following article, the field of scalars denoted F is either the field of real numbers R or the field of complex numbers C. See below.

Formally, an inner product space is a vector space V over the field F together with a bilinear form, called an inner product

$\langle \cdot ,\cdot \rangle :V\times V\rightarrow \mathbf {F}$

satisfying the following axioms:

• Nonnegativity:

$\forall x\in V,\ \langle x,x\rangle \geq 0.$

• Nondegeneracy:

$\forall x\in V,\ \langle x,x\rangle =0{\mbox{ iff }}x=0.$

• Conjugate symmetry:

$\forall x,y\in V,\ \langle x,y\rangle ={\overline {\langle y,x\rangle }}$

(Conjugation is also often written with an asterisk, as in <y,x>*, as is the conjugate transpose.)

• Sesquilinearity:

$\forall b\in F,\ \forall x,y\in V,\ \langle x,by\rangle =b\langle x,y\rangle$

$\forall x,y,z\in V,\ \langle x,y+z\rangle =\langle x,y\rangle +\langle x,z\rangle .$

By combining these with conjugate symmetry, we get:

$\forall a\in F,\ \forall x,y\in V,\ \langle ax,y\rangle ={\overline {a}}\langle x,y\rangle$

$\forall x,y,z\in V,\ \langle x+y,z\rangle =\langle x,z\rangle +\langle y,z\rangle .$

Note that if F=R, then the conjugate symmetry property is simply symmetry of the inner product, i.e.

$\langle x,y\rangle =\langle y,x\rangle$

In this case, sesquilinearity becomes standard linearity.

Remark. Many mathematical authors require an inner product to be linear in the first argument and conjugate-linear in the second argument, contrary to the convention adopted above. This change is immaterial, but the definition above ensures a smoother connection to the bra-ket notation used by physicists in quantum mechanics and is now often used by mathematicians as well. Some authors adopt the convention that < , > is linear in the first component while < | > is linear in the second component, although this is by no means universal. For instance the G. Emch reference does not follow this convention.

There are various technical reasons why we have to restrict the basefield to R and C in the definition. Briefly the basefield has to contain an ordered subfield (in order for non-negativity to make sense) and therefore has to have characteristic equal to 0. This immediately excludes finite fields. The basefield has to have additional structure, such as a distinguished automorphism.

In some cases we need to consider non-negative semi-definite sesquilinear forms. This means that <x, x> is only required to be non-negative. We show how to treat these below.

Examples

A trivial example are the real numbers with the standard multiplication as the inner product

$\langle x,y\rangle :=xy$

More generally any Euclidean space Rⁿ with the dot product is an inner product space

$\langle (x_{1},\ldots ,x_{n}),(y_{1},\ldots ,y_{n})\rangle :=\sum _{i=1}^{n}x_{i}y_{i}=x_{1}y_{1}+\cdots +x_{n}y_{n}$

Even more generally any positive-definite matrix M can be used to define an inner product on Cⁿ as

$\langle \mathbf {x} ,\mathbf {y} \rangle :=\mathbf {x} ^{*}\mathbf {M} \mathbf {y}$

with x^* the conjugate transpose of x.

The article on Hilbert space has several examples of inner product spaces where the metric induced by the inner product yields a complete metric spaces. An example of an inner product which induces an incomplete metric is the space C[a, b] of continuous complex valued functions on the interval [a,b]. The inner product is

$\langle f,g\rangle :=\int _{a}^{b}{\overline {f(t)}}g(t)\,dt$

This space is not complete; consider for example, for the interval [0,1] the sequence of functions { f_k }_k where

• f_k(t) is 1 for t in the subinterval [0, 1/2]
• f_k(t) is 0 for t in the subinterval [1/2 + 1/k, 1]
• f_k is affine in [1/2, 1/2 + 1/k]

This sequence is a Cauchy sequence which does not converge to a continuous function.

Norms on inner product spaces

Inner product spaces have a naturally defined norm

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

This is well defined by the nonnegatity axiom of the definition of inner product space. The norm is thought of as the length of the vector x. Directly from the axioms, we can prove the following:

• Cauchy-Schwarz inequality: for x, y elements of V

$|\langle x,y\rangle |\leq \|x\|\cdot \|y\|$

with equality if and only if x and y are linearly dependent. This is one of the most important inequalities in mathematics. It is also known in the Russian mathematical literature as the Cauchy-Bunyakowski-Schwarz inequality.

Because of its importance, its short proof should be noted. To prove this inequality note it is trivial in the case y = 0. Thus we may assume <y, y> is nonzero. Thus we may let

$\lambda =\langle y,y\rangle ^{-1}\langle x,y\rangle$

and it follows that

$0\leq \langle x-\lambda y,x-\lambda y\rangle =\langle x,x\rangle -\langle y,y\rangle ^{-1}|\langle x,y\rangle |^{2}.$

multiplying out, the result follows.

Geometric interpretation of inner product

The geometric interpretation of the inner product in terms of angle and length, motivates much of the geometric terminology we use in regard to these spaces. Indeed, an immediate consequence of the Cauchy-Schwarz inequality is that it justifies defining the angle between two non-zero vectors x and y (at least in the case F = R) by the identity

$\operatorname {angle} (x,y)=\arccos {\frac {\langle x,y\rangle }{\|x\|\cdot \|y\|}}.$

We assume the value of the angle is chosen to be in the interval (−π, +π]. This is in analogy to the familiar situation in two-dimensional Euclidean space. Correspondingly, we will say that non-zero vectors x, y of V are orthogonal iff their inner product is zero.

• Homogeneity: for x an element of V an r a scalar

$\|r\cdot x\|=|r|\cdot \|x\|.$

The homogeneity property is completely trivial to prove.

• Triangle inequality: for x, y elements of V

$\|x+y\|\leq \|x\|+\|y\|.$

The last two properties show the function defined is indeed a norm.

Because of the triangle inequality and because of axiom 2, we see that ||·|| is a norm which turns V into a normed vector space and hence also into a metric space. The most important inner product spaces are the ones which are complete with respect to this metric; they are called Hilbert spaces. Every inner product V space is a dense subspace of some Hilbert space. This Hilbert space is essentially uniquely determined by V and is constructed by completing V.

• Parallelogram law:

$\|x+y\|^{2}+\|x-y\|^{2}=2\|x\|^{2}+2\|y\|^{2}.$

• Pythagorean theorem: Whenever x, y are in V and <x, y> = 0, then

$\|x\|^{2}+\|y\|^{2}=\|x+y\|^{2}.$

The proofs of both of these identities require only expressing the definition of norm in terms of the inner product and multiplying out, using the property of additivity of each component. The name Pythagorean theorem arises from the geometric interpretation of this result as an analogue of the theorem in synthetic geometry. Note that the proof of the Pythagorean theorem in synthetic geometry is considerably more elaborate because of the paucity of underlying structure. In this sense, the synthetic Pythagorean theorem, if correctly demonstrated is deeper than the version given above.

An easy induction on the Pythagorean theorem yields:

• If x₁, ..., x_n are orthogonal vectors, that is, <x_j, x_k> = 0 for distinct indices j, k, then

$\sum _{i=1}^{n}\|x_{i}\|^{2}=\left\|\sum _{i=1}^{n}x_{i}\right\|^{2}.$

In view of the Cauchy-Schwarz inequality, we also note that <·,·> is continuous from V × V to F. This allows us to extend Pythagoras' theorem to infinitely many summands:

• Parseval's identity: Suppose V is a complete inner product space. If {x_k} are mutually orthogonal vectors in V then

$\sum _{i=1}^{\infty }\|x_{i}\|^{2}=\left\|\sum _{i=1}^{\infty }x_{i}\right\|^{2},$

provided the infinite series on the left is convergent. Completeness of the space is needed to insure that the sequence of partial sums

$S_{k}=\sum _{i=1}^{k}x_{i}$

which is easily shown to be a Cauchy sequence is convergent.

Orthonormal sequences

A sequence {e_k}_k is orthonormal iff it is orthogonal and each e_k has norm 1. An orthornormal basis for an inner product space V is an orthonormal sequence whose algebraic span is V.

The Gram-Schmidt process is a canonical procedure that takes a linearly independent sequence {v_k}_k on an inner product space and produces an orthonormal sequence {e_k}_k such that for each n

$\operatorname {span} \{v_{1},\ldots ,v_{n}\}=\operatorname {span} \{e_{1},\ldots ,e_{n}\}$

By the Gram-Schmidt orthonormalization process, one shows:

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

Parseval's identity leads immediately to the following theorem:

Theorem. Let V be a separable inner product space and {e_k}_k an orthonormal basis of V. Then the map

$x\mapsto \{\langle e_{k},x\rangle \}_{k\in \mathbb {N} }$

is an isometric linear map V → l² with a dense image.

This theorem can be regarded as an abstract form of Fourier series, in which an arbitrary orthornormal 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 l² 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[-π,π]. Then the sequence (indexed on set of all integers) of continuous functions

$e_{k}(t)=(2\pi )^{-1/2}e^{ikt}$

is an orthonormal basis of the space C[-π,π] with the L² inner product. The mapping

$f\mapsto {\frac {1}{\sqrt {2\pi }}}\left\{\int _{-\pi }^{\pi }f(t)e^{-ikt}\,dt\right\}_{k\in \mathbb {Z} }$

is an isometric linear map with dense image.

Orthogonality of the sequence {e_k}_k follows immediately from the fact that if k ≠ j, then

$\int _{-\pi }^{\pi }e^{-i(j-k)t}\,dt=0$

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 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 [-π, π] with the uniform norm. This is the content of the Weierstrauss theorem on the uniform density of trigonometric polynomials.

Operators on inner product spaces

Several types of linear maps A from an inner product space V to an inner product space W are of relevance:

• Continuous linear maps, i.e. A is linear and continuous with respect to the metric defined above, or equivalently, A is linear and the set of non-negative reals {||Ax||}, where x ranges over the closed unit ball of V, is bounded.
• Symmetric linear operators, i.e. A is linear and <Ax, y> = <x, A y> for all x, y in V.
• Isometries, i.e. A is linear and <Ax, Ay> = <x, y> for all x, y in V, or equivalently, A is linear and ||Ax|| = ||x|| for all x in V. All isometries are injective.
• Isometrical isomorphisms, i.e. A 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.

Degenerate inner products

If V is a vector space and < , > a semi-definite sesquilinear form, then the function ||x|| = <x,x>^1/2 makes sense and satisfies all the properties of norm except that ||x|| = 0 does not imply x = 0. We can produce an inner product space by considering the quotient W = V/{x:||x|| = 0}. The sesquilinear form < , > 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.

References

• S. Axler, Linear Algebra Done Right, Springer, 2004

• G. Emch, Algebraic Methods in Statistical Mechanics and Quantum Field Theory, Wiley Interscience, 1972.

• N. Young, An Introduction to Hilbert Spaces, Cambridge University Press, 1988

See also

• outer product