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 ?
The contributors describe distance functions for reproducing kernel Hilbert spaces, the properties of locally m-pseudoconvex algebras, spectral preservers in commutative Banach spaces, and peak point theorems for uniform algebras on manifolds.
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Participants of the July 2008 conference share recent research on affine transformation crossed product type algebras and noncommutative surfaces, C*-algebras associated with iterated function systems, extending representations of normed algebras in Banach spaces, and freeness of group actions on C*-algebras.
Fixed point equations and nonlinear eigen value problems in ordered Banach spaces, SIAM, Rev.