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 It might be outdated or ideologically biased.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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References in periodicals archive ?
Let L be a Banach function space (briefly, B.f.s.) over some positive measure space ([OMEGA], [summation], [mu]), X be a Banach space and T : L [right arrow] X be a continuous linear operator.
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.
Recall that a Banach function space X is p-convex if there is a constant [K.sub.p] such that for every finite set [f.sub.1], ..., [f.sub.n] [member of] X, the inequality
Following the definition in [16, p.28], a Banach space X([mu]) of (classes of) locally [mu]-integrable real functions is said to be a Banach function space over [mu] (Kothe function space) if it satisfies the next two properties.
Then [[rho].sub.p[*],[delta](*)] is a Banach function norm, and the associated Banach function space
From now on we assume that (E, [[parallel] x [parallel].sub.E]) is a Banach function space. Then the space E(X) provided with the norm [[parallel] f [parallel].sub.E(X)] := [[parallel] [??] [parallel].sub.E] is a Banach space and is usually called a Kothe-Bochner function space.
Such a space X is called a Banach function space. If [rho] is a function norm, its associate norm p' is defined on [L.sup.0.sub.+] by
Their topics include the measurability and semi-continuity of multifunctions, the optimality of function spaces in Sobolev embeddings, a note on the off-diagonal Muckenhoupt-Wheedon conjecture, the Radon-Nikod<'y>m theorem for vector measures and integral representation of operators on Banach function spaces, and the Orlicz-Pettis theorem for multiplier convergent series.
The Hardy inequalities on the Morrey spaces built on rearrangement-invariant Banach function spaces are obtained [13].
For the case of Banach function spaces, Theorems 2 and 5 were obtained in [7], and then in [3] they were generalized for the case of Banach lattices in some other view.
Perez, "The Hardy-Littlewood maximal type operators between Banach function spaces," Indiana University Mathematics Journal, vol.
The problem of integral representation of bounded linear operators on Banach function spaces of vector-valued functions to Banach spaces in terms of the corresponding operator-valued measures has been the object of much study (see [5, 17-24]).