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 ?
Let C be a nonempty closed convex subset of a real Banach space X and T : C [right] C a nearly uniformly L-Lipschitzian mapping with sequence {[r.
n]) in a Banach space is said to be orthogonal if [parallel][summation][a.
1]([0,1],R) the Banach space of all continuously differentiable functions from [0, 1] into R with the norm
Kaiser in (14) proved the stability of monomial functional equation where the functions map a normed space over a field with valuation to a Banach space over a field with valuation and the control function is of the form [epsilon]([||x||.
p] is a uniformly bounded subset of the Banach Space.
Let (X, [parallel]*[parallel]) be a Banach space and let B(X) be the Banach algebra of all linear and bounded operators acting from X into X.
where F is a Frechet-differential operator defined on an open convex domain D of a Banach space X with values in a Banach space Y.
Let (V; +, x) be a Banach space over a field F and [?
59-60]) that a norm [parallel]x[parallel] of a Banach space X is uniformly convex if its modulus of convexity [delta]X : (0,2] [right arrow] [0,1], defined by
His topics include how the embeddability of locally finite metric spaces into Banach space is finitely determined, constructions of embeddings, Banach spaces that do not admit uniformly coarse embeddings of expanders, applying Markov chains to embeddability problems, and Lipschitz free spaces.
Many authors have extended, generalized and improved the answer to Ulam's question such as, Hyers (4) in the context of Banach space, K.