In linear algebra, an inner product space or a Hausdorff pre-Hilbert space^{[1]}^{[2]} 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, often denoted using angle brackets (as in ).^{[3]} 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,^{[4]} also known as the scalar product) to vector spaces of any (possibly infinite) dimension, and are studied 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.^{[5]}
An inner product naturally induces an associated norm, (|x| and |y| are the norms of x and y, in the picture), which canonically makes every inner product space into a normed vector space. If this normed space is also a Banach space then the inner product space is called a Hilbert space.^{[1]} If an inner product space (H, ⟨·, ·⟩) is not a Hilbert space then it can be "extended" to a Hilbert space (H, ⟨·, ·⟩_{H}), called a completion. Explicitly, this means that H is linearly and isometrically embedded onto a dense vector subspace of H and that the inner product ⟨·, ·⟩_{H} on H is the unique continuous extension of the original inner product ⟨·, ·⟩.^{[1]}^{[6]}
In this article, the field of scalars denoted F is either the field of real numbers R or the field of complex numbers C.
Formally, an inner product space is a vector space V over the field F together with a map
called an inner product that satisfies the following conditions (1), (2), and (3)^{[1]} for all vectors x, y, z ? V and all scalars a ? F:^{[7]}^{[8]}^{[9]}
The above three conditions are the defining properties of an inner product, which is why an inner product is sometimes (equivalently) defined as being a positive-definite Hermitian form. An inner product can equivalently be defined as a positive-definite sesquilinear form.^{[1]}^{[note 5]}
Assuming (1) holds, condition (3) will hold if and only if both conditions (4) and (5) below hold:^{[6]}^{[1]}
Positive-definiteness and linearity, respectively, ensure that:
Notice that conjugate symmetry implies that ⟨x, x⟩ is real for all x, 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.
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 F = R, conjugate-symmetry reduces to symmetry, and sesquilinearity reduces to bilinearity. Hence an inner product on a real vector space is a positive-definite symmetric bilinear form. That is,
and the binomial expansion becomes:
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 ⟨x, y⟩ as (the bra-ket notation of quantum mechanics), respectively y^{+}x (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 linear functionals (covectors) of the dual space V^{*}, with conjugacy associated with duality. This reverse order is now occasionally followed in the more abstract literature,^{[10]} taking ⟨x, y⟩ 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 base field to R and C in the definition. Briefly, the base field has to contain an ordered subfield in order for non-negativity to make sense,^{[11]} 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 R or C will suffice for this purpose (e.g., algebraic numbers, constructible numbers). However, in the cases where 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, one needs to consider non-negative semi-definite sesquilinear forms. This means that ⟨x, x⟩ is only required to be non-negative. Treatment for these cases are illustrated below.
A simple example is the real numbers with the standard multiplication as the inner product^{[4]}
More generally, the real n-space R^{n} with the dot product is an inner product space,^{[4]} an example of a Euclidean vector space.
where x^{T} is the transpose of x.
The general form of an inner product on C^{n} is known as the Hermitian form and is given by
where M is any Hermitian positive-definite matrix and y^{+} 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.
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
This space is not complete; consider for example, for the interval [-1, 1] the sequence of continuous "step" functions, { 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.
For real random variables X and Y, the expected value of their product
is an inner product.^{[12]}^{[13]}^{[14]} In this case, ⟨X, X⟩ = 0 if and only if Pr(X = 0) = 1 (i.e., X = 0 almost surely). This definition of expectation as inner product can be extended to random vectors as well.
For real square matrices of the same size, ⟨A, B⟩ := tr(AB^{T}) with transpose as conjugation
is an inner product.
On an inner product space, or more generally a vector space with a nondegenerate form (hence an isomorphism V -> 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.
Inner product spaces are normed vector spaces for the norm defined by^{[4]}
As for every normed vector space, a inner product space is a metric space, for the distance defined by
The axioms of the inner product guarantee that the map above forms a norm, which will have the following properties.
Suppose that is an inner product on V (so it is antilinear in its second argument). The polarization identity shows that the real part of the inner product is
If is a real vector space then and the imaginary part (also called the complex part) of is always 0.
Assume for the rest of this section that V is a complex vector space. The polarization identity for complex vector spaces shows that
The map defined by for all satisfies the axioms of the inner product except that it is antilinear in its first, rather than its second, argument. The real part of both and are equal to but the inner products differ in their complex part:
The last equality is similar to the formula expressing a linear functional in terms of its real part.
Let denote considered as a vector space over the real, rather than complex, numbers. The map is an inner product on the real vector space Every inner product on a real vector space is symmetric and bilinear.
For example, if with inner product where is a vector space over the field then is a vector space over and is the dot product where is identified with the point (and similarly for ). Also, had been instead defined to be the symmetric (rather than antisymmetric) map then its real part would not be the dot product.
The next examples show that although real and complex inner products have many properties and results in common, they are not entirely interchangeable. For instance, if then but the next example shows that the converse is in general not true. Given any , the vector (which is the vector rotated by 90°) belongs to and so also belongs to (although scalar multiplication of by i is not defined in , it is still true that the vector denoted by is an element of ). For the complex inner product, whereas for the real inner product the value is always
If has the inner product mentioned above, then the map defined by is a non-zero linear map (linear for both and ) that denotes rotation by 90° in the plane. This map satisfies for all vectors where had this inner product been complex instead of real, then this would have been enough to conclude that this linear map is identically (i.e. that ), which rotation is certainly not. In contrast, for all non-zero , the map satisfies .
Let V be a finite dimensional inner product space of dimension n. Recall that every basis of V consists of exactly 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 {e_{1}, ..., e_{n}} is orthonormal if ⟨e_{i}, e_{j}⟩ = 0 for every i ? j and ⟨e_{i}, e_{i}⟩ = |||| = 1 for each i.
This definition of orthonormal basis generalizes to the case of infinite-dimensional inner product spaces in the following way. Let V be any inner product space. Then a collection
is a basis for V if the subspace of V generated by finite linear combinations of elements of E is dense in V (in the norm induced by the inner product). We say that E is an orthonormal basis for V if it is a basis and
if ? ? ? and ⟨e_{?}, e_{?}⟩ = |||| = 1 for all ?, ? ? A.
Using an infinite-dimensional analog of the Gram-Schmidt process one may show:
Theorem. Any separable inner product space 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 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).^{[]}
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
is an isometric linear map V -> l^{2} 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 l^{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 V be the inner product space C[-?, ?]. Then the sequence (indexed on set of all integers) of continuous functions
is an orthonormal basis of the space C[-?, ?] with the L^{2} inner product. The mapping
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
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.
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.
If V is a vector space and ⟨·, ·⟩ a semi-definite sesquilinear form, then the function:
makes sense and satisfies all the properties of norm except that |||| = 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 : |||| = 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.
Alternatively, one may require that the pairing be a nondegenerate form, meaning that for all non-zero x there exists some y such that ⟨x, y⟩ ? 0, though y need not equal x; in other words, the induced map to the dual space V -> V^{*} 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 V -> V^{*}) and thus hold more generally.
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 × n covector with an n × 1 vector, yielding a 1 × 1 matrix (a scalar), while the outer product is the product of an m × 1 vector with a 1 × n covector, yielding an m × n 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 (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 × V^{*} -> 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^{*} × V -> 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).