Inner product space
The inner product of two vectors in the space is a scalar, often denoted with angle brackets such as in
Inner product spaces of infinite dimension are widely used in functional analysis.
The first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in 1898.
Hence an inner product on a real vector space is a positive-definite symmetric bilinear form.
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.
It is a weighted-sum version of the dot product with positive weights—up to an orthogonal transformation.
This space is not complete; consider for example, for the interval [−1, 1] the sequence of continuous "step" functions,
This definition of expectation as inner product can be extended to random vectors as well.
Since trace and transposition are linear and the conjugation is on the second matrix, it is a sesquilinear operator.
The polarization identity shows that the real part of the inner product is
The polarization identity for complex vector spaces shows that The map defined by
These formulas show that every complex inner product is completely determined by its real part.
is considered as a real vector space in the usual way (meaning that it is identified with the
Every inner product on a real vector space is a bilinear and symmetric map.
The next examples show that although real and complex inner products have many properties and results in common, they are not entirely interchangeable.
This definition of orthonormal basis generalizes to the case of infinite-dimensional inner product spaces in the following way.
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.
The two previous theorems raise the question of whether all inner product spaces have an orthonormal basis.
The following proof is taken from Halmos's A Hilbert Space Problem Book (see the references).
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.
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
This is the content of the Weierstrass theorem on the uniform density of trigonometric polynomials.
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.
The generalizations that are closest to inner products occur where bilinearity and conjugate symmetry are retained, but positive-definiteness is weakened.
Alternatively, one may require that the pairing be a nondegenerate form, meaning that for all non-zero
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.
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
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
