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In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space.Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space.
In functional analysis, the type and cotype of a Banach space are a classification of Banach spaces through probability theory and a measure, how far a Banach space from a Hilbert space is. The starting point is the Pythagorean identity for orthogonal vectors ( e k ) k = 1 n {\displaystyle (e_{k})_{k=1}^{n}} in Hilbert spaces
There is an obvious algebraic duality between the vector space of all finitely additive measures σ on Σ and the vector space of simple functions (() = ()). It is easy to check that the linear form induced by σ is continuous in the sup-norm if σ is bounded, and the result follows since a linear form on the dense subspace of simple functions ...
Tsirelson space, a reflexive Banach space in which neither nor can be embedded. W.T. Gowers construction of a space X {\displaystyle X} that is isomorphic to X ⊕ X ⊕ X {\displaystyle X\oplus X\oplus X} but not X ⊕ X {\displaystyle X\oplus X} serves as a counterexample for weakening the premises of the Schroeder–Bernstein theorem [ 1 ]
Uniform Boundedness Principle — Let be a Banach space, a normed vector space and (,) the space of all continuous linear operators from into . Suppose that F {\displaystyle F} is a collection of continuous linear operators from X {\displaystyle X} to Y . {\displaystyle Y.}
The construction of a Banach space without the approximation property earned Per Enflo a live goose in 1972, which had been promised by Stanisław Mazur (left) in 1936. [1] In mathematics, specifically functional analysis, a Banach space is said to have the approximation property (AP), if every compact operator is a limit of finite-rank ...
Importantly, this theorem fails for complex Banach spaces. [2] However, for the special case where B {\displaystyle B} is the closed unit ball then this theorem does hold for complex Banach spaces. [ 1 ] [ 2 ]
Many standard Banach spaces have this property, most notably, the space () of continuous functions on a compact space and the space () of the Lebesgue integrable functions on a measure space. Alexander Grothendieck introduced the concept in the early 1950s ( Grothendieck 1953 ), following the work of Dunford and Pettis, who developed earlier ...