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  2. Reflexive space - Wikipedia

    en.wikipedia.org/wiki/Reflexive_space

    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 ...

  3. Banach space - Wikipedia

    en.wikipedia.org/wiki/Banach_space

    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 ...

  4. List of Banach spaces - Wikipedia

    en.wikipedia.org/wiki/List_of_Banach_spaces

    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 ]

  5. Uniformly convex space - Wikipedia

    en.wikipedia.org/wiki/Uniformly_convex_space

    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 ...

  6. James's theorem - Wikipedia

    en.wikipedia.org/wiki/James's_theorem

    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 ...

  7. Compact operator - Wikipedia

    en.wikipedia.org/wiki/Compact_operator

    If X is a reflexive Banach space, then every completely continuous operator T : X → Y is compact. Somewhat confusingly, compact operators are sometimes referred to as "completely continuous" in older literature, even though they are not necessarily completely continuous by the definition of that phrase in modern terminology.

  8. Polynomially reflexive space - Wikipedia

    en.wikipedia.org/wiki/Polynomially_reflexive_space

    In mathematics, a polynomially reflexive space is a Banach space X, on which the space of all polynomials in each degree is a reflexive space. Given a multilinear functional M n of degree n (that is, M n is n -linear), we can define a polynomial p as

  9. Riesz's lemma - Wikipedia

    en.wikipedia.org/wiki/Riesz's_lemma

    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.