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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 ...
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.}
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
In mathematics, an Orlicz sequence space is any of certain class of linear spaces of scalar-valued sequences, endowed with a special norm, specified below, under which it forms a Banach space. Orlicz sequence spaces generalize the ℓ p {\displaystyle \ell _{p}} spaces, and as such play an important role in functional analysis .
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 X {\displaystyle X} is a locally compact Hausdorff space.
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If Y is a Banach space, an equivalent definition is that the embedding operator (the identity) i : X → Y is a compact operator. When applied to functional analysis, this version of compact embedding is usually used with Banach spaces of functions. Several of the Sobolev embedding theorems are compact embedding theorems.