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 ?
m[greater than or equal to]0] in the Banach space [mathematical expression not reproducible].
On the other hand, in [17] it was shown that for a real Banach space X and a compact Hausdorff space K, if T : X [right arrow] [C.
Amann, "On the number of solutions of nonlinear equations in ordered Banach space," Journal of Functional Analysis, vol.
Mann [11] defined a more general iteration in a Banach space E satisfying quasi-nonexpansive operators.
Triggiani: Controllability and observability in Banach space with bounded operators, SIAM J.
It is based on recent breakthrough results on the representation of sets as functions in a Banach space, and will allow the design of high performance numerical schemes for problems in dynamical systems and control theory, which cannot be solved efficiently within the traditional framework.
Next covered are metrics, Banach space, construction of topological vector spaces, map and graph theorems, local convexity, and an assortment of additional topics deemed necessary to proceed to chapter 8, which addresses duality.
1, we equip N with a weighted counting measure, and let V be the corresponding Banach space of p-power summable sequences.
t[greater than or equal to]0] is defined on a complex Banach space E, f : [0, T] x B x E [right arrow] E is a given function.
16] Suppose [perpendicular to] is symmetric on L and X is Banach space.
to obtain a strong convergence theorem for uniformly Lipschitzian asymptotically pseudocontractive mapping in real Banach space setting.