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  2. Banach space - Wikipedia

    en.wikipedia.org/wiki/Banach_space

    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.

  3. Type and cotype of a Banach space - Wikipedia

    en.wikipedia.org/wiki/Type_and_cotype_of_a...

    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

  4. ba space - Wikipedia

    en.wikipedia.org/wiki/Ba_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 ...

  5. Banach fixed-point theorem - Wikipedia

    en.wikipedia.org/wiki/Banach_fixed-point_theorem

    In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem or Banach–Caccioppoli theorem) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces and provides a constructive method to find those fixed points.

  6. Uniform boundedness principle - Wikipedia

    en.wikipedia.org/wiki/Uniform_boundedness_principle

    Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerstones of the field. In its basic form, it asserts that for a family of continuous linear operators (and thus bounded operators) whose domain is a Banach space, pointwise boundedness is equivalent to uniform boundedness in operator norm.

  7. Open mapping theorem (functional analysis) - Wikipedia

    en.wikipedia.org/wiki/Open_mapping_theorem...

    In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem or the Banach theorem [1] (named after Stefan Banach and Juliusz Schauder), is a fundamental result that states that if a bounded or continuous linear operator between Banach spaces is surjective then it is an open map.

  8. 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 ]

  9. Schur's property - Wikipedia

    en.wikipedia.org/wiki/Schur's_property

    Download as PDF; Printable version ... When we are working in a normed space X and we have a ... Robert E. (1998), An Introduction to Banach Space Theory, New York ...