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...

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....

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...

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...

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...

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...

functional analysis, Banach spaces are among the most important objects of study. In other areas of mathematical analysis, most spaces which arise in practice...

that bear Banach's name include Banach spaces, Banach algebras, Banach measures, the Banach–Tarski paradox, the Hahn–Banach theorem, the Banach–Steinhaus...

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...

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...

{\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...

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...

Banach. (Heinonen 2003) Every separable metric space is isometric to a subset of the Urysohn universal space. For nonseparable spaces: A metric space...

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...

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...

Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces (normed vector spaces that...

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...

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...

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...

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...

mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem or Banach–Caccioppoli theorem)...

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...

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...

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...

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...

Fréchet spaces are locally convex spaces that are completely metrizable (with a choice of complete metric). They are generalizations of Banach spaces, which...

approximation problem and later the invariant subspace problem for Banach spaces. In solving these problems, Enflo developed new techniques which were...

operation of direct sum in finite-dimensional vector spaces. Every finite-dimensional subspace of a Banach space is complemented, but other subspaces may not...

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...

In mathematics, a Banach manifold is a manifold modeled on Banach spaces. Thus it is a topological space in which each point has a neighbourhood homeomorphic...