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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.
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 ]
It is a closed linear subspace of the space of bounded sequences, , and contains as a closed subspace the Banach space of sequences converging to zero. The dual of c {\displaystyle c} is isometrically isomorphic to ℓ 1 , {\displaystyle \ell ^{1},} as is that of c 0 . {\displaystyle c_{0}.}
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 space c 0 is defined as the space of all sequences converging to zero, with norm identical to ||x|| ∞. It is a closed subspace of ℓ ∞, hence a Banach space. The dual of c 0 is ℓ 1; the dual of ℓ 1 is ℓ ∞. For the case of natural numbers index set, the ℓ p and c 0 are separable, with the sole exception of ℓ ∞.
This is a Banach space (in fact a commutative Banach algebra with identity) with respect to the uniform norm. (Hewitt & Stromberg 1965, Theorem 7.9) It is sometimes desirable, particularly in measure theory, to further refine this general definition by considering the special case when is a locally compact Hausdorff
An infinite-dimensional Banach space is called indecomposable whenever its only complemented subspaces are either finite-dimensional or -codimensional. Because a finite- codimensional subspace of a Banach space X {\\displaystyle X} is always isomorphic to X , {\\displaystyle X,} indecomposable Banach spaces are prime.
If X is a Banach space with a Schauder basis {e n} n ≥ 1 such that the biorthogonal functionals are a basis of the dual, that is to say, a Banach space with a shrinking basis, then the space K(X) admits a basis formed by the rank one operators e* j ⊗ e k : v → e* j (v) e k, with the same ordering as before. [17]