Inner product space


In linear algebra, an inner product space is a vector space with an additional structure called an 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. They also provide the means of defining orthogonality between vectors. Inner product spaces generalize Euclidean spaces to vector spaces of any dimension, and are studied in functional analysis. The first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in 1898.
An inner product naturally induces an associated norm, thus an inner product space is also a normed vector space. A complete space with an inner product is called a Hilbert space. An space with an inner product is called a pre-Hilbert space, since its completion with respect to the norm induced by the inner product is a Hilbert space. Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces.

Definition

In this article, the field of scalars denoted is either the field of real numbers or the field of complex numbers.
Formally, an inner product space is a vector space over the field together with an inner product, i.e., with a map
that satisfies the following three properties for all vectors and all scalars :
If the positive-definite condition is replaced by merely requiring that for all x, then one obtains the definition of positive semi-definite Hermitian form. A positive semi-definite Hermitian form is an inner product if and only if for all x, if then x = 0.

Elementary properties

Positive-definiteness and linearity, respectively, ensure that:
Notice that conjugate symmetry implies that is real for all, since we have:
Conjugate symmetry and linearity in the first variable imply
that is, conjugate linearity in the second argument. So, an inner product is a sesquilinear form. Conjugate symmetry is also called Hermitian symmetry, and a conjugate-symmetric sesquilinear form is called a Hermitian form. While the above axioms are more mathematically economical, a compact verbal definition of an inner product is a positive-definite Hermitian form.
This important generalization of the familiar square expansion follows:
These properties, constituents of the above linearity in the first and second argument:
are otherwise known as additivity.
In the case of, conjugate-symmetry reduces to symmetry, and sesquilinearity reduces to bilinearity. So, an inner product on a real vector space is a positive-definite symmetric bilinear form. That is,
and the binomial expansion becomes:

Alternative definitions, notations and remarks

A common special case of the inner product, the scalar product or dot product, is written with a centered dot.
Some authors, especially in physics and matrix algebra, prefer to define the inner product and the sesquilinear form with linearity in the second argument rather than the first. Then the first argument becomes conjugate linear, rather than the second. In those disciplines we would write the product as , respectively . Here the kets and columns are identified with the vectors of and the bras and rows with the linear functionals of the dual space, with conjugacy associated with duality. This reverse order is now occasionally followed in the more abstract literature, taking to be conjugate linear in rather than. 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 base field to and in the definition. Briefly, the base field 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 or will suffice for this purpose, e.g., the algebraic numbers or the constructible numbers. However, in these cases when it is a proper subfield even finite-dimensional inner product spaces will fail to be metrically complete. In contrast all finite-dimensional inner product spaces over or, 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.

Some examples

Real numbers

A simple example is the real numbers with the standard multiplication as the inner product

Euclidean vector space

More generally, the real -space with the dot product is an inner product space, an example of a Euclidean vector space.
where is the transpose of.

Complex coordinate space

The general form of an inner product on is known as the Hermitian form and is given by
where is any Hermitian positive-definite matrix and is the conjugate transpose of. 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. Up to an orthogonal transformation it is a weighted-sum version of the dot product, with positive weights.

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 which induces an incomplete metric occurs with the space of continuous complex valued functions f and g on the interval. The inner product is
This space is not complete; consider for example, for the interval the sequence of continuous "step" functions,, defined by:
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

For real random variables and, the expected value of their product
is an inner product. In this case, if and only if . This definition of expectation as inner product can be extended to random vectors as well.

Real matrices

For real square matrices of the same size, with transpose as conjugation
is an inner product.

Vector spaces with forms

On an inner product space, or more generally a vector space with a nondegenerate form vectors can be sent to covectors, so one can take the inner product and outer product of two vectors, not simply of a vector and a covector.

Norm

Inner product spaces are normed vector spaces for the norm defined by
As for every normed vector space, a inner product space is a metric space, for the distance defined by
Directly from the axioms of the inner product, one can prove that the axioms of a norm are satisfied, as well as the following properties.

Orthonormal sequences

Let be a finite dimensional inner product space of dimension. Recall that every basis of consists of exactly 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 is orthonormal if for every and for each.
This definition of orthonormal basis generalizes to the case of infinite-dimensional inner product spaces in the following way. Let be any inner product space. Then a collection
is a basis for if the subspace of generated by finite linear combinations of elements of is dense in . We say that is an orthonormal basis for if it is a basis and
if and for all.
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.
Let be a Hilbert space of dimension . Let be an orthonormal basis of, so. Extend to a Hamel basis for, where. Since it is known that the Hamel dimension of is, the cardinality of the continuum, it must be that.
Let be a Hilbert space of dimension . Let be an orthonormal basis for, and let be a bijection. Then there is a linear transformation such that for, and for.
Let and let be the graph of. Let be the closure of in ; we will show. Since for any we have, it follows that.
Next, if, then for some, so ; since as well, we also have. It follows that, so, and is dense in.
Finally, is a maximal orthonormal set in ; if
for all then certainly, so is the zero vector in. Hence the dimension of is, whereas it is clear that the dimension of is. This completes the proof.
Parseval's identity leads immediately to the following theorem:
Theorem. Let be a separable inner product space and an orthonormal basis of . Then the map
is an isometric linear map 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. In particular, we obtain the following result in the theory of Fourier series:
Theorem. Let be the inner product space. Then the sequence of continuous functions
is an orthonormal basis of the space with the inner product. The mapping
is an isometric linear map with dense image.
Orthogonality of the sequence follows immediately from the fact that if, 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.

Operators on inner product spaces

Several types of linear maps from an inner product space to an inner product space 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.

Generalizations

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.

Degenerate inner products

If is a vector space and a semi-definite sesquilinear form, then the function:
makes sense and satisfies all the properties of norm except that does not imply . We can produce an inner product space by considering the quotient. The sesquilinear form factors through.
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.

Nondegenerate conjugate symmetric forms

Alternatively, one may require that the pairing be a nondegenerate form, meaning that for all non-zero there exists some such that, though need not equal ; in other words, the induced map to the dual space 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 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.
Purely algebraic statements usually only rely on the nondegeneracy and thus hold more generally.

Related products

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 covector with an vector, yielding a 1 × 1 matrix, while the outer product is the product of an vector with a covector, yielding an matrix. Note that 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. 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 sending a vector and a covector to a rank 1 linear transformation, while the inner product is the bilinear evaluation map 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 product are combined in the geometric product – the inner product sends two vectors to a scalar, while the exterior product sends two vectors to a bivector – and in this context the exterior product is usually called the outer 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.