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In mathematics, more specifically in functional analysis, a Banach space (/ ˈ b ɑː. n ʌ x /, Polish pronunciation:) 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 ...
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}.}
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.
Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerstones of the field. In its basic form, it asserts that for a family of continuous linear operators (and thus bounded operators) whose domain is a Banach space, pointwise boundedness is equivalent to uniform boundedness in operator norm.
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach space, that is, a normed space that is complete in the metric induced by the norm.