Search results
Results From The WOW.Com Content Network
A Banach space is reflexive if it is linearly isometric to its bidual under this canonical embedding . James' space is an example of a non-reflexive space which is linearly isometric to its bidual. Furthermore, the image of James' space under the canonical embedding J {\displaystyle J} has codimension one in its bidual.
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 ]
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 ...
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.}
The space T* is a minimal Banach space. [9] This means that every infinite-dimensional Banach subspace of T* contains a further subspace isomorphic to T*. Prior to the construction of T*, the only known examples of minimal spaces were ℓ p and c 0. The dual space T is not minimal. [10] The space T* is polynomially reflexive.
Every non-reflexive infinite-dimensional Banach space is a distinguished space that is not semi-reflexive. [11] If is a dense proper vector subspace of a reflexive Banach space then is a normed space that not semi-reflexive but its strong dual space is a reflexive Banach space. [11]
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).