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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 ...
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 ]
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 open mapping theorem, also known as the Banach–Schauder theorem or the Banach theorem [1] (named after Stefan Banach and Juliusz Schauder), is a fundamental result that states that if a bounded or continuous linear operator between Banach spaces is surjective then it is an open map.
In functional analysis, an important class of vector spaces consists of the complete normed vector spaces over the real or complex numbers, which are called Banach spaces. An important example of a Banach space is a Hilbert space , where the norm arises from an inner product .
The topological dual of -Banach space deduced from by any restriction scalar will be denoted ′. (It is of interest only if is a complex space because if is a -space then ′ = ′. James compactness criterion — Let X {\displaystyle X} be a Banach space and A {\displaystyle A} a weakly closed nonempty subset of X . {\displaystyle X.}