Geometric interpretation of the angle between two vectors defined using an inner product
Scalar product spaces, over any field, have "scalar products" that are symmetrical and linear in the first argument. Hermitian product spaces are restricted to the field of complex numbers and have "Hermitian products" that are conjugate-symmetrical and linear in the first argument. Inner product spaces may be defined over any field, having "inner products" that are linear in the first argument, conjugate-symmetrical, and positive-definite. Unlike inner products, scalar products and Hermitian products need not be positive-definite.
In this article, F denotes a field that is either the real numbers or the complex numbers A scalar is thus an element of F. A bar over an expression representing a scalar denotes the complex conjugate of this scalar. A zero vector is denoted for distinguishing it from the scalar 0.
An inner product space is a vector spaceV over the field F together with an inner product, that is a map
that satisfies the following three properties for all vectors and all scalars .
As if and only if a is real, conjugate symmetry implies that is always a real number. If F is , conjugate symmetry is just symmetry.
If the positive-definiteness 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.
Over , 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. The binomial expansion of a square becomes
Some authors, especially in physics and matrix algebra, prefer to define inner products and sesquilinear forms with linearity in the second argument rather than the first. Then the first argument becomes conjugate linear, rather than the second.
A function is an inner product on if and only if there exists a symmetricpositive-definite matrix such that for all If is the identity matrix then is the dot product. For another example, if and is positive-definite (which happens if and only if and one/both diagonal elements are positive) then for any
As mentioned earlier, every inner product on is of this form (where and satisfy ).
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 of continuous complex valued functions and on the interval The inner product is
This space is not complete; consider for example, for the interval [−1, 1] 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.
The inner product for complex square matrices of the same size is the Frobenius inner product. Since trace and transposition are linear and the conjugation is on the second matrix, it is a sesquilinear operator. We further get Hermitian symmetry by,
Finally, since for nonzero, , we get that the Frobenius inner product is positive definite too, and so is an inner product.
On an inner product space, or more generally a vector space with a nondegenerate form (hence an isomorphism ), 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.
Two vectors and are said to be orthogonal, often written if their inner product is zero, that is, if
This happens if and only if for all scalars  and if and only if the real-valued function is non-negative. (This is a consequence of the fact that, if then the scalar minimizes with value which is always non positive).
For a complex − but not real[clarification needed] − inner product space a linear operator is identically if and only if for every 
This may be proved by expressing the squared norms in terms of the inner products, using additivity for expanding the right-hand side of the equation.
The name Pythagorean theorem arises from the geometric interpretation in Euclidean geometry.
Assume for the rest of this section that 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:
These formulas show that every complex inner product is completely determined by its real part. Moreover, this real part defines an inner product on considered as a real vector space. There is thus a one-to-one correspondence between complex inner products on a complex vector space and real inner products on
For example, suppose that for some integer When is considered as a real vector space in the usual way (meaning that it is identified with the dimensional real vector space with each identified with ), then the dot product defines a real inner product on this space. The unique complex inner product on induced by the dot product is the map that sends to (because the real part of this map is equal to the dot product).
Real vs. complex inner products
Let denote considered as a vector space over the real numbers rather than complex numbers.
The real part of the complex inner product is the map which necessarily forms a real inner product on the real vector space Every inner product on a real vector space is a bilinear and symmetric map.
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 ); thus the standard inner product on is an "extension" the dot product . Also, had been instead defined to be the symmetric map (rather than the usual conjugate symmetric map) then its real part would not be the dot product; furthermore, without the complex conjugate, if but then so the assignment would not define a norm.
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 is not defined in the vector in denoted by is nevertheless still also an element of ). For the complex inner product, whereas for the real inner product the value is always
If is a complex inner product and is a continuous linear operator that satisfies for all then This statement is no longer true if is instead a real inner product, as this next example shows.
Suppose that has the inner product mentioned above. Then the map defined by is a linear map (linear for both and ) that denotes rotation by in the plane. Because and perpendicular vectors and is just the dot product, for all vectors nevertheless, this rotation map is certainly not identically In contrast, using the complex inner product gives which (as expected) is not identically zero.
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 index
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 (in the norm induced by the inner product). 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.
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).
Recall that the dimension of an inner product space is the cardinality of a maximal orthonormal system that it contains (by Zorn's lemma it contains at least one, and any two have the same cardinality). An orthonormal basis is certainly a maximal orthonormal system but the converse need not hold in general. If is a dense subspace of an inner product space then any orthonormal basis for is automatically an orthonormal basis for Thus, it suffices to construct an inner product space with a dense subspace whose dimension is strictly smaller than that of
Let be a Hilbert space of dimension (for instance, ). 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 (for instance, ). 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 so is the zero vector in Hence the dimension of is whereas it is clear that the dimension of is This completes the proof.
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 (and in fact any set whatsoever, provided 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 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 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.
Several types of linear maps between inner product spaces and are of relevance:
Continuous linear maps: is linear and continuous with respect to the metric defined above, or equivalently, is linear and the set of non-negative reals where ranges over the closed unit ball of is bounded.
Symmetric linear operators: is linear and for all
Isometries: satisfies for all A linear isometry (resp. an antilinear isometry) is an isometry that is also a linear map (resp. an antilinear map). For inner product spaces, the polarization identity can be used to show that is an isometry if and only if for all All isometries are injective. The Mazur–Ulam theorem establishes that every surjective isometry between two real normed spaces is an affine transformation. Consequently, an isometry between real inner product spaces is a linear map if and only if Isometries are morphisms between inner product spaces, and morphisms of real inner product spaces are orthogonal transformations (compare with orthogonal matrix).
Isometrical isomorphisms: 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.
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 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 (such a functional is then called a semi-norm). We can produce an inner product space by considering the quotient The sesquilinear form factors through
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 (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 ) 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 covector with an vector, yielding a matrix (a scalar), while the outer product is the product of an vector with a covector, yielding an matrix. 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 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 given by evaluating a covector on a vector; the order of the domain vector spaces here reflects the covector/vector distinction.
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).
^By combining the linear in the first argument property with the conjugate symmetry property you get conjugate-linear in the second argument: . This is how the inner product was originally defined and is used in most mathematical contexts. A different convention has been adopted in theoretical physics and quantum mechanics, originating in the bra-ket notation of Paul Dirac, where the inner product is taken to be linear in the second argument and conjugate-linear in the first argument; this convention is used in many other domains such as engineering and computer science.