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 ?
Their approach is to reduce the Cauchy problems of PDE's to that of abstract ordinary differential equations on a scale of Banach spaces.
Both of them are Banach spaces with the supremum norm.
Among the highlights are spectral, structural, and geometric properties of special types of operators on Banach spaces, emphasizing isometries, weighted composition operators, and projections on function spaces.
For stating the result, we need the following lemma concerning smooth points of the spheres in Banach spaces.
From 2008, the authors have some results with regard to the approximation algorithm, the approximation solution for a variety of generalized nonlinear ordered variational inequalities, ordered equations and inclusions, and sensitivity analysis for a class of parametric variational inclusions in ordered Banach spaces (see [7-22]).
Zhang, "The convergence of implicit Mann and Ishikawa iterations for weak generalized [phi]-hemicontractive mappings in real Banach spaces," Journal of Inequalities and Applications, vol.
Metric embeddings; bilipschitz and coarse embeddings into Banach spaces.
Motivated by these works, we show that a particular class of impulsive fractional differential systems in Banach spaces is controllable provided that some conditions have to be satisfied.
The importance for applications consists in the richness of the structure of modular spaces, that-besides being Banach spaces (or F-spaces in more general setting)- are equipped with modular equivalent of norm or metric notions.
Kang, Iterative approximation of fixed points and solutions for strongly accretive and strongly pseudo-contractive mappings in Banach spaces, J.