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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 ...
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 a non-reflexive Banach space, such as the Lebesgue space () of all bounded sequences, Riesz’s lemma does not hold for =. [ 5 ] However, every finite dimensional normed space is a reflexive Banach space, so Riesz’s lemma does holds for α = 1 {\displaystyle \alpha =1} when the normed space is finite-dimensional, as will now be shown.
B-convex space; Ba space; Banach bundle (non-commutative geometry) Banach bundle; Banach manifold; List of Banach spaces; Besov space; Bishop–Phelps theorem; BK-space; Browder–Minty theorem; Bs space