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.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Banach Space


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

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.

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].
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Discussing stability problems of certain classes of hybrid and shock systems, Martynyuk, Radziszewsky, and Szadkowski cover hybrid systems on a time scale with behavior described by dynamic equations; hybrid systems with after-effects under pulse perturbations; hybrid weakly-coupled systems with subsystems defined in the Banach spaces; mechanical systems with impact described by the Poincare mapping; the bouncing-ball model expressed by discrete mapping and difference equations; and common-recurrence equations and inequalities, linearization techniques, and global estimates.
Kirk, "On successive approximations for non-expansive mappings in Banach spaces," Glasgow Mathematical Journal, vol.
In this paper, we present some sufficient conditions which ensure the existence of solution to BVP (4) in Banach spaces. Through construction space DC(I), using the technique of the Kuratowski noncompactness measure and the Sadovskii fixed point theorem, we obtain some new existence criteria for BVP (4).
In recent years, fractional differential equations in Banach spaces have been studied and a few papers consider fractional differential equations in reflexive Banach spaces equipped with the weak topology.
Silvestri, "Integral equalities for functions of unbounded spectral operators in Banach spaces," Dissertationes Mathematicae, no.
For the autonomous systems, Phat and Kiet [8] investigated relationship between stability and exact null controllability extending the Lyapunov equation in Banach spaces. The smart characterization of generator of the perturbation semigroup for Pritchard-Salamon systems was provided by Guo et al.
Best approximative properties of hyperplanes or subspaces in general, of Banach spaces are closely related to structure of the unit ball and its exposed faces and study of geometric structure of the unit ball in view of this link, is not new [2].
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Myjak, Weak convergence of convex sets in Banach spaces, Arch.
Existence and uniqueness of the solution of second-order nonlinear systems and controllability of these systems in Banach spaces have been studied thoroughly by many authors [1,2, 4, 12, 13].
Several issues emphasized by Daleckii and Krein at the 5th International Congress of Nonlinear Oscillations, held in Kiev in 1969, have focused on the necessity of the study of asymptotic behaviors of differential equations with bounded coefficients on Banach spaces. This important step in the development of the stability theory consists in a natural launch in the study of evolution equations given by unbounded operators.