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 It might be outdated or ideologically biased.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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The results about the BVP with multiple point boundary conditions of impulsive p-Laplacian operator fractional differential equations are few, especially in Banach space.
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Recall that the right-hand quotient Ao[B.sup.-1] of two operator ideals A and B is the operator ideal that consists of all operators T [member of] L(X,Y) such that TS [member of] A ([X.sub.0],Y) whenever S [member of] B([X.SUB.0],X) for some Banach space [X.sub.0] (see [6,3.1.1]).
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Where [mathematical expression not reproducible] and f : J x E x E x [OMEGA] [right arrow] E are given continuous functions, ([OMEGA], A, v) is a measurable space, and E is a real (or complex) Banach space with norm [[parallel] * [parallel].sub.e] and dual [E.sup.*], such that E is the dual of a weakly compactly generated Banach space [mathematical expression not reproducible], is the left-sided mixed Hadamard integral of order r, and [mathematical expression not reproducible] is a given continuous and measurable function such that