HILBERT SPACES

 

Hilbert spaces[edit]

Main article: Hilbert space

The succeeding snapshots show summation of 1 to 5 terms in approximating a periodic function (blue) by finite sum of sine functions (red).

Complete inner product spaces are known as Hilbert spaces, in honor of David Hilbert.^[51] The Hilbert space ${\displaystyle {�}^2(\mathrm{\Omega }),}$ with inner product given by

$${\displaystyle ⟨�,�⟩={\int }_{\mathrm{\Omega }}�(�)\overline{�(�)}��,}$$

where $\overline{�(�)}$ denotes the complex conjugate of $�(�),$^{[52][nb 11]} is a key case.

By definition, in a Hilbert space any Cauchy sequence converges to a limit. Conversely, finding a sequence of functions ${�}_{�}$ with desirable properties that approximates a given limit function, is equally crucial. Early analysis, in the guise of the Taylor approximation, established an approximation of differentiable functions $�$ by polynomials.^[53] By the Stone–Weierstrass theorem, every continuous function on $[�,�]$ can be approximated as closely as desired by a polynomial.^[54] A similar approximation technique by trigonometric functions is commonly called Fourier expansion, and is much applied in engineering, see below.^{[clarification needed]} 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 $�,$ 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 $�,$ its cardinality is known as the Hilbert space dimension.^{[nb 12]} 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.^[55] 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.^[56] 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.^[57] 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.^[58]