Search results
Results From The WOW.Com Content Network
A Banach space is super-reflexive if all Banach spaces finitely representable in are reflexive, or, in other words, if no non-reflexive space is finitely representable in . The notion of ultraproduct of a family of Banach spaces [ 14 ] allows for a concise definition: the Banach space X {\displaystyle X} is super-reflexive when its ultrapowers ...
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 ...
In social theory, reflexivity may occur when theories in a discipline should apply equally to the discipline itself; for example, in the case that the theories of knowledge construction in the field of sociology of scientific knowledge should apply equally to knowledge construction by sociology of scientific knowledge practitioners, or when the subject matter of a discipline should apply ...
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 ]
The unit sphere can be replaced with the closed unit ball in the definition. Namely, a normed vector space is uniformly convex if and only if for every < there is some > so that, for any two vectors and in the closed unit ball (i.e. ‖ ‖ and ‖ ‖) with ‖ ‖, one has ‖ + ‖ (note that, given , the corresponding value of could be smaller than the one provided by the original weaker ...
In 1957, James had proved the reflexivity criterion for separable Banach spaces [2] and 1964 for general Banach spaces. [3] Since the reflexivity is equivalent to the weak compactness of the unit sphere, Victor L. Klee reformulated this as a compactness criterion for the unit sphere in 1962 and assumes that this criterion characterizes any ...
In mathematics, the Milman–Pettis theorem states that every uniformly convex Banach space is reflexive.. The theorem was proved independently by D. Milman (1938) and B. J. Pettis (1939).
Robert C. James characterized reflexivity in Banach spaces with basis: the space X with a Schauder basis is reflexive if and only if the basis is both shrinking and boundedly complete. [24] James also proved that a space with an unconditional basis is non-reflexive if and only if it contains a subspace isomorphic to c 0 or ℓ 1. [25]