In mathematics, an inner product space is a vector space with the additional structure of inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors. Inner products allow the rigorous introduction of intuitive geometrical notions such as the length of a vector or the angle between two vectors. It also provides the means of defining orthogonality between vectors (zero scalar product). Inner product spaces generalize Euclidean spaces (in which the inner product is the dot product, also known as the scalar product) to vector spaces of any (possibly infinite) dimension, and are studied in functional analysis.
An inner product space is sometimes also called a pre-Hilbert space, since its completion with respect to the metric, induced by its inner product, is a Hilbert space.
Inner product spaces were referred to as unitary spaces in earlier work, although this terminology is now rarely used.
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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 positive-definite sesquilinear form, called an inner product. For real vector spaces, this is actually a positive-definite symmetric bilinear form. Thus the inner product is a map
satisfying the following axioms for all :
Remark: In the more abstract linear algebra literature, the conjugate-linear argument of the inner product is conventionally put in the second position (e.g., y in ) as we have done above. This convention is reversed in both physics and matrix algebra; in those respective disciplines we would write the product
as
(the bra-ket notation of quantum mechanics) and
(dot product as a case of the convention of forming the matrix product AB as the dot products of rows of A with columns of B). Here the kets and columns are identified with the vectors of V and the bras and rows with the dual vectors or linear functionals of the dual space V*, with conjugacy associated with duality. This reverse order is now occasionally followed in the more abstract literature, e.g., Emch [1972], taking
to be conjugate linear in x rather than y. A few instead find a middle ground by recognizing both
and
as distinct notations differing only in which argument is conjugate linear.
There are various technical reasons why it is necessary 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. More generally any quadratically closed subfield of R or C will suffice for this purpose, e.g., the algebraic numbers, but when it is a proper subfield (i.e., neither R nor C) even finite-dimensional inner product spaces will fail to be metrically complete. In contrast all finite-dimensional inner product spaces over R or C, such as those used in quantum computation, are automatically metrically complete and hence Hilbert spaces.
In some cases we need to consider non-negative semi-definite sesquilinear forms. This means that is only required to be non-negative. We show how to treat these below.
A trivial example is the real numbers with the standard multiplication as the inner product
More generally any Euclidean space Rn with the dot product is an inner product space
The general form of an inner product on Cn is given by:
with M any symmetric positive-definite matrix, and y* the conjugate transpose of y. For the real case this corresponds to the dot product of the results of directionally differential scaling of the two vectors, with positive scale factors and orthogonal directions of scaling. Apart from an orthogonal transformation it is a weighted-sum version of the dot product, with positive weights.
The article on Hilbert space 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 which induces an incomplete metric occurs with the space C[a, b] of continuous complex valued functions on the interval [a, b]. The inner product is
This space is not complete; consider for example, for the interval [−1,1] the sequence of "step" functions { fk }k where
This sequence is a Cauchy sequence which does not converge to a continuous function.
Inner product spaces have a naturally defined norm
This is well defined by the nonnegativity 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:
A sequence {ek}k is orthonormal if and only if it is orthogonal and each ek has norm 1. An orthonormal basis for an inner product space of finite dimension V is an orthonormal sequence whose algebraic span is V. This definition of orthonormal basis does not generalise conveniently to the case of infinite dimensions, where the concept (properly formulated) is of major importance. Using the norm associated to the inner product, one has the notion of dense subset, and the appropriate definition of orthonormal basis is that the algebraic span (subspace of finite linear combinations of basis vectors) should be dense.
The Gram-Schmidt process is a canonical procedure that takes a linearly independent sequence {vk}k on an inner product space and produces an orthonormal sequence {ek}k such that for each 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 {ek}k an orthonormal basis of V. Then the map
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 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 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 . Then the sequence (indexed on set of all integers) of continuous functions
is an orthonormal basis of the space with the L2 inner product. The mapping
is an isometric linear map with dense image.
Orthogonality of the sequence {ek}k follows immediately from the fact that if k ≠ j, then
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 with the uniform norm. This is the content of the Weierstrass theorem on the uniform density of trigonometric polynomials.
Several types of linear maps A from an inner product space V to an inner product space 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.
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. (Such a functional is then called a semi-norm.) 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.
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