functional analysis, a Banach space (pronounced [ˈbanax]) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows...
103 KB (17,216 words) - 04:41, 23 July 2024
and the normed space is a Banach space. Those spaces for which the canonical evaluation map is surjective are called semi-reflexive spaces. In 1951, R....
39 KB (6,405 words) - 08:22, 19 April 2024
complete normed field) that at the same time is also a Banach space, that is, a normed space that is complete in the metric induced by the norm. The...
17 KB (2,602 words) - 13:01, 17 July 2024
vector space can be "uniquely extended" to a Banach space, which makes normed spaces intimately related to Banach spaces. Every Banach space is a normed...
18 KB (2,890 words) - 22:11, 21 February 2024
the space is a complete metric space. A Hilbert space is a special case of a Banach space. The earliest Hilbert spaces were studied from this point of...
128 KB (17,487 words) - 22:34, 20 June 2024
to make the study of the dual space "interesting". Another version of the Hahn–Banach theorem is known as the Hahn–Banach separation theorem or the hyperplane...
77 KB (12,643 words) - 19:21, 19 April 2024
functional analysis, Banach spaces are among the most important objects of study. In other areas of mathematical analysis, most spaces which arise in practice...
2 KB (450 words) - 23:44, 26 July 2024
that bear Banach's name include Banach spaces, Banach algebras, Banach measures, the Banach–Tarski paradox, the Hahn–Banach theorem, the Banach–Steinhaus...
26 KB (2,699 words) - 18:24, 23 February 2024
The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in three-dimensional space, there exists...
48 KB (6,821 words) - 19:51, 26 July 2024
Direct sum of modules (redirect from Direct sum of Banach spaces)
spaces (modules over a field) and abelian groups (modules over the ring Z of integers). The construction may also be extended to cover Banach spaces and...
22 KB (3,560 words) - 16:29, 14 May 2024
{\displaystyle \ell ^{p}} is a complete metric space with respect to this norm, and therefore is a Banach space. If p = 2 {\displaystyle p=2} then ℓ 2 {\displaystyle...
22 KB (3,603 words) - 16:12, 23 February 2024
Banach. (Heinonen 2003) Every separable metric space is isometric to a subset of the Urysohn universal space. For nonseparable spaces: A metric space...
14 KB (2,071 words) - 12:06, 2 March 2024
is a Banach space, and so a complete metric space, with respect to the supremum norm. However, the supremum norm does not give a norm on the space C(a...
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quotient space W/im(T). If X is a Banach space and M is a closed subspace of X, then the quotient X/M is again a Banach space. The quotient space is already...
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Riesz (Riesz 1910). Lp spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces. Because of their key role...
69 KB (12,904 words) - 15:52, 18 July 2024
Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces (normed vector spaces that...
29 KB (5,029 words) - 22:25, 5 April 2024
Functional analysis (section Banach spaces)
non-negative integers. In Banach spaces, a large part of the study involves the dual space: the space of all continuous linear maps from the space into its underlying...
20 KB (2,499 words) - 18:10, 28 July 2024
the Tsirelson space is the first example of a Banach space in which neither an ℓ p space nor a c0 space can be embedded. The Tsirelson space is reflexive...
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adjoint space [Alaoglu 1940], and transponierter Raum [Schauder 1930] and [Banach 1932]. The term dual is due to Bourbaki 1938. Given any vector space V {\displaystyle...
45 KB (6,872 words) - 18:21, 24 June 2024
linear space endowed with a norm is a normed space. Every normed space is both a linear topological space and a metric space. A Banach space is a complete...
69 KB (9,311 words) - 05:27, 20 May 2024
mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem or Banach–Caccioppoli theorem)...
16 KB (2,529 words) - 15:35, 7 June 2024
mathematics, the Banach–Alaoglu theorem (also known as Alaoglu's theorem) states that the closed unit ball of the dual space of a normed vector space is compact...
61 KB (8,306 words) - 08:04, 6 March 2024
Open mapping theorem (functional analysis) (redirect from Banach-Schauder theorem)
that states that if a bounded or continuous linear operator between Banach spaces is surjective then it is an open map. A special case is also called...
21 KB (3,958 words) - 07:09, 26 July 2024
L-infinity (redirect from L-infinity-space)
former is a special case of the latter. As a Banach space they are the continuous dual of the Banach spaces ℓ 1 {\displaystyle \ell _{1}} of absolutely...
5 KB (770 words) - 12:24, 25 June 2024
in a suitable weak sense to make the space complete, i.e. a Banach space. Intuitively, a Sobolev space is a space of functions possessing sufficiently...
35 KB (6,406 words) - 06:53, 30 May 2024
line, any separable Banach space, the Cantor space, and the Baire space. Additionally, some spaces that are not complete metric spaces in the usual metric...
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Complemented subspace (redirect from Indecomposable Banach space)
operation of direct sum in finite-dimensional vector spaces. Every finite-dimensional subspace of a Banach space is complemented, but other subspaces may not...
21 KB (3,308 words) - 08:12, 11 April 2024
topological vector space is not a Banach space, then there is a good chance that it is nuclear. Much of the theory of nuclear spaces was developed by Alexander...
27 KB (4,344 words) - 16:00, 8 May 2024
Schauder basis (category Banach spaces)
for the analysis of infinite-dimensional topological vector spaces including Banach spaces. Schauder bases were described by Juliusz Schauder in 1927,...
26 KB (3,590 words) - 10:05, 12 January 2024
Fréchet spaces are locally convex spaces that are completely metrizable (with a choice of complete metric). They are generalizations of Banach spaces, which...
58 KB (10,602 words) - 00:35, 10 July 2024