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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.
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 ...
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
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 ]
A bounded disk in a topological vector space such that (,) is a Banach space is called a Banach disk, infracomplete, or a bounded completant in . If its shown that ( span D , p D ) {\displaystyle \left(\operatorname {span} D,p_{D}\right)} is a Banach space then D {\displaystyle D} will be a Banach disk in any TVS that contains D ...
Let (X, ‖·‖) be a Banach space over a field K (either the real or complex numbers), and let Ext(X) be the set of extreme points of the closed unit ball of the continuous dual space X ∗. A continuous linear operator T : X → X is said to be a multiplier if every point p in Ext(X) is an eigenvector for the adjoint operator T ∗ : X ∗ ...
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