Editing Vector space (section)

====Hilbert spaces====
{{Main|Hilbert space}}
[[Image:Periodic identity function.gif|class=skin-invert-image|right|thumb|400px|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]].{{sfn|Treves|1967|loc=ch. 12}} The Hilbert space <math>L^2(\Omega),</math> with inner product given by
<math display=block>\langle f\ , \ g \rangle = \int_\Omega f(x) \overline{g(x)} \, dx,</math>
where <math>\overline{g(x)}</math> denotes the [[complex conjugate]] of <math>g(x),</math>{{sfn|Dennery|Krzywicki|1996|loc = p.190}}<ref group=nb>For <math>p \neq 2,</math> <math>L^p(\Omega)</math> is not a Hilbert space.</ref> is a key case.

By definition, in a Hilbert space, any Cauchy sequence converges to a limit. Conversely, finding a sequence of functions <math>f_n</math> 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 function]]s <math>f</math> by polynomials.{{sfn|Lang|1993|loc = Th. XIII.6, p. 349}} By the [[Stone–Weierstrass theorem]], every continuous function on <math>[a, b]</math> can be approximated as closely as desired by a polynomial.{{sfn|Lang|1993|loc = Th. III.1.1}} A similar approximation technique by [[trigonometric function]]s 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 <math>H,</math> in the sense that the ''[[closure (topology)|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 <math>H,</math> its cardinality is known as the [[Hilbert space dimension]].<ref group=nb>A basis of a Hilbert space is not the same thing as a basis of a linear algebra. For distinction, a linear algebra basis for a Hilbert space is called a [[Hamel basis]].</ref> 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 [[orthogonal basis|basis of orthogonal vectors]].{{sfn|Choquet|1966|loc = Lemma III.16.11}} Such orthogonal bases are the Hilbert space generalization of the coordinate axes in finite-dimensional [[Euclidean space]].

The solutions to various [[differential equation]]s 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.{{sfn|Kreyszig|1999|loc=Chapter 11}} 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 [[wavefunction]]s.{{sfn|Griffiths|1995|loc=Chapter 1}} Definite values for physical properties such as energy, or momentum, correspond to [[eigenvalue]]s of a certain (linear) [[differential operator]] and the associated wavefunctions are called [[eigenstate]]s. The <span id=labelSpectralTheorem>[[spectral theorem]] decomposes a linear [[compact operator]] acting on functions in terms of these eigenfunctions and their eigenvalues.</span>{{sfn|Lang|1993|loc =ch. XVII.3}}