“Inner product space”的意思、由来-开放百科全书

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 (zero inner 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. The first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in 1898.^[1]

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 (incomplete) 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 {{math|F}} is either the field of real numbers {{math|R}} or the field of complex numbers {{math|C}}.

Formally, an inner product space is a vector space {{math|V}} over the field {{math|F}} together with an inner product, i.e., with a map

that satisfies the following three axioms for all vectors {{math|x, y, z ∈ V}} and all scalars {{math|a ∈ F}}:^[2]^[3]

Elementary properties

When {{math|F {{=}} R}}, conjugate symmetry reduces to symmetry. That is, {{math|{{angbr|x,y}} {{=}} {{angbr|y,x}}}} for {{math|F {{=}} R}}; while for {{math|F {{=}} C}}, {{math|{{angbr|x,y}}}} is equal to the complex conjugate.

Notice that conjugate symmetry implies that {{math|{{angbr|x,x}}}} is real for all {{math|x}}, since we have:

so an inner product is a sesquilinear form, in the sense that the second argument is conjugate linear. 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.

In the case of {{math|F {{=}} R}}, conjugate-symmetry reduces to symmetry, and sesquilinear reduces to bilinear. So, an inner product on a real vector space is a positive-definite symmetric bilinear form.

Combining the linearity of the inner product in its first argument and the conjugate symmetry gives the following important generalization of the familiar square expansion:

Assuming the underlying field to be {{math|R}}, the inner product becomes symmetric, and we obtain

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 {{math|{{angbr|x,y}}}} as {{math|{{bra-ket|y|x}}}} (the bra–ket notation of quantum mechanics), respectively {{math|y^†x}} (dot product as a case of the convention of forming the matrix product {{math|AB}} as the dot products of rows of {{math|A}} with columns of {{math|B}}). Here the kets and columns are identified with the vectors of {{math|V}} and the bras and rows with the linear functionals (covectors) of the dual space {{math|V{{isup|∗}}}}, with conjugacy associated with duality. This reverse order is now occasionally followed in the more abstract literature,^[5] taking {{math|{{angbr|x,y}}}} to be conjugate linear in {{math|x}} rather than {{math|y}}. A few instead find a middle ground by recognizing both {{math|{{angbr|·,·}}}} and {{math|{{bra-ket|·|·}}}} 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 {{math|R}} and {{math|C}} in the definition. Briefly, the basefield has to contain an ordered subfield in order for non-negativity to make sense,^[6] and therefore has to have characteristic equal to 0 (since any ordered field has to have such characteristic). This immediately excludes finite fields. The basefield has to have additional structure, such as a distinguished automorphism. More generally any quadratically closed subfield of {{math|R}} or {{math|C}} will suffice for this purpose, e.g., the algebraic numbers or the constructible numbers. However in these cases when it is a proper subfield (i.e., neither {{math|R}} nor {{math|C}}) even finite-dimensional inner product spaces will fail to be metrically complete. In contrast all finite-dimensional inner product spaces over {{math|R}} or {{math|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 {{math|{{angbr|x,x}}}} is only required to be non-negative. We show how to treat these below.

Examples

Real numbers

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

Euclidean space

More generally, the real {{math|n}}-space {{math|Rⁿ}} with the dot product is an inner product space, an example of a Euclidean {{math|n}}-space.

Complex coordinate space

The general form of an inner product on {{math|Cⁿ}} is known as the Hermitian form and is given by

where {{math|M}} is any Hermitian positive-definite matrix and {{math|y^†}} is the conjugate transpose of {{math|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. Up to an orthogonal transformation it is a weighted-sum version of the dot product, with positive weights.

Hilbert space

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 {{math|C([a,b])}} of continuous complex valued functions f and g on the interval {{math|[a,b]}}. The inner product is

This space is not complete; consider for example, for the interval {{closed-closed|−1,1}} the sequence of continuous "step" functions, {{math|{ f_k}_k}}, 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 {{math|X}} and {{math|Y}}, the expected value of their product

is an inner product.^[7]^[8]^[9] In this case, {{math|{{angbr|X,X}} {{=}} 0}} if and only if {{math|Pr(X {{=}} 0) {{=}} 1}} (i.e., {{math|X {{=}} 0}} almost surely). 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, {{math|{{angbr|A,B}} :{{=}} tr(AB^T)}} with transpose as conjugation

Vector spaces with forms

On an inner product space, or more generally a vector space with a nondegenerate form (so an isomorphism {{math|V → V{{isup|∗}}}}) vectors can be sent to covectors (in coordinates, via transpose), so one can take the inner product and outer product of two vectors, not simply of a vector and a covector.

is a normed space but not an inner product space, because this norm does not satisfy the parallelogram equality required of a norm to have an inner product associated with it.^[10]^[11]

However, inner product spaces have a naturally defined norm based upon the inner product of the space itself that does satisfy the parallelogram equality:

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 {{math|x}}. Directly from the axioms, we can prove the following:

Orthonormal sequences

Let {{math|V}} be a finite dimensional inner product space of dimension {{math|n}}. Recall that every basis of {{math|V}} consists of exactly {{math|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 {{math|{e₁, ..., e_n}}} is orthonormal if {{math|{{angbr|e_i,e_j}} {{=}} 0}} for every {{math|i ≠ j}} and {{math|{{angbr|e_i,e_i}} {{=}} {{norm|e_i}} {{=}} 1}} for each {{math|i}}.

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

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

if {{math|α ≠ β}} and {{math|{{angbr|e_α,e_α}} {{=}} {{norm|e_α}} {{=}} 1}} for all {{math|α, β ∈ A}}.

Theorem. Any separable inner product space {{math|V}} 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 {{math|V}} 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}}

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

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 {{math|l{{isup|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 {{math|V}} be the inner product space {{math|C[−π,π]}}. Then the sequence (indexed on set of all integers) of continuous functions

is an orthonormal basis of the space {{math|C[−π,π]}} with the {{math|L²}} inner product. The mapping

Orthogonality of the sequence {{math|{e_k}_k}} follows immediately from the fact that if {{math|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 {{math|[−π,π]}} 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 {{math|A}} from an inner product space {{math|V}} to an inner product space {{math|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.

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 {{math|V}} is a vector space and {{math|{{angbr|·,·}}}} a semi-definite sesquilinear form, then the function:

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

Nondegenerate conjugate symmetric forms

Alternatively, one may require that the pairing be a nondegenerate form, meaning that for all non-zero {{math|x}} there exists some {{math|y}} such that {{math|{{angbr|x,y}} ≠ 0}}, though {{math|y}} need not equal {{math|x}}; in other words, the induced map to the dual space {{math|V → V{{isup|∗}}}} 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 (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 {{math|V → V{{isup|∗}}}}) and thus hold more generally.