information:extraction
Hilbert spaces

Complete inner product spaces are known as Hilbert spaces, in honor of David Hilbert. The Hilbert space L 2 (Ω), {\displaystyle L^{2}(\Omega),} with inner product given by ⟨ f , g ⟩ = ∫ Ω f (x) g (x) ¯ d x, {\displaystyle \langle f\,\ g\rangle =\int _{\Omega }f(x){\overline {g(x)}}\,dx,} where g (x) ¯ {\displaystyle {\overline {g(x)}}} denotes the complex conjugate of g (x), {\displaystyle g(x),} is a key case.

By definition, in a Hilbert space, any Cauchy sequence converges to a limit. Conversely, finding a sequence of functions f n {\displaystyle f_{n}} with desirable properties that approximate a given limit function is equally crucial. Early analysis, in the guise of the Taylor approximation, established an approximation of differentiable functions f {\displaystyle f} by polynomials. By the Stone–Weierstrass theorem, every continuous function on {\displaystyle } can be approximated as closely as desired by a polynomial. A similar approximation technique by trigonometric functions is commonly called Fourier expansion, and is much applied in engineering. More generally, and more conceptually, the theorem yields a simple description of what "basic functions", or, in abstract Hilbert spaces, what basic vectors suffice to generate a Hilbert space H, {\displaystyle H,} in the sense that the closure of their span (that is, finite linear combinations and limits of those) is the whole space. Such a set of functions is called a basis of H, {\displaystyle H,} its cardinality is known as the Hilbert space dimension. Not only does the theorem exhibit suitable basis functions as sufficient for approximation purposes, but also together with the Gram–Schmidt process, it enables one to construct a basis of orthogonal vectors. Such orthogonal bases are the Hilbert space generalization of the coordinate axes in finite-dimensional Euclidean space.

The solutions to various differential equations can be interpreted in terms of Hilbert spaces. For example, a great many fields in physics and engineering lead to such equations, and frequently solutions with particular physical properties are used as basis functions, often orthogonal. As an example from physics, the time-dependent Schrödinger equation in quantum mechanics describes the change of physical properties in time by means of a partial differential equation, whose solutions are called wavefunctions. Definite values for physical properties such as energy, or momentum, correspond to eigenvalues of a certain (linear) differential operator and the associated wavefunctions are called eigenstates. The spectral theorem decomposes a linear compact operator acting on functions in terms of these eigenfunctions and their eigenvalues.