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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 a canonical evaluation map ′ (,) (from the projective tensor product of and to the Banach space of continuous linear maps from to ). It is determined by sending f ∈ A ′ {\displaystyle f\in A^{\prime }} and b ∈ B {\displaystyle b\in B} to the linear map a ↦ f ( a ) ⋅ b . {\displaystyle a\mapsto f(a)\cdot b.}
A Banach space with a Schauder basis is necessarily separable, but the converse is false. Since every vector v in a Banach space V with a Schauder basis is the limit of P n (v), with P n of finite rank and uniformly bounded, such a space V satisfies the bounded approximation property.
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 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
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.}
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