Banach space

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Banach space

[′bä‚näk ‚spās]
A real or complex vector space in which each vector has a non-negative length, or norm, and in which every Cauchy sequence converges to a point of the space. Also known as complete normed linear space.

Banach Space


(named after S. Banach), a complete normed linear space.

Banach space

A complete normed vector space. Metric is induced by the norm: d(x,y) = ||x-y||. Completeness means that every Cauchy sequence converges to an element of the space. All finite-dimensional real and complex normed vector spaces are complete and thus are Banach spaces.

Using absolute value for the norm, the real numbers are a Banach space whereas the rationals are not. This is because there are sequences of rationals that converges to irrationals.

Several theorems hold only in Banach spaces, e.g. the Banach inverse mapping theorem. All finite-dimensional real and complex vector spaces are Banach spaces. Hilbert spaces, spaces of integrable functions, and spaces of absolutely convergent series are examples of infinite-dimensional Banach spaces. Applications include wavelets, signal processing, and radar.

[Robert E. Megginson, "An Introduction to Banach Space Theory", Graduate Texts in Mathematics, 183, Springer Verlag, September 1998].
References in periodicals archive ?
Specific topics include Young-Fenchel transformation and some new characteristics of Banach spaces, an example of the boundary of topologically inverted elements, sums and products of bad functions, disc algebra and a moment problem, the stability of logmodularity for uniform algebras, regularity and amenability conditions for uniform algebras, closed suns of marginal subspaces of Banach function space, surjections on the algebras of continuous functions which preserve peripheral spectrum, asymptotics of Toeplitz determinants generated by functions with Fourier coefficients in weighted Orlicz sequence classes, spectral isometries, examples of Banach spaces that are not Banach algebras, and a spectra of algebras of analytic functions and polynomials on Banach space.