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2. Dunford–Pettis property - Wikipedia

   en.wikipedia.org/wiki/Dunford–Pettis_property

   Many standard Banach spaces have this property, most notably, the space () of continuous functions on a compact space and the space () of the Lebesgue integrable functions on a measure space. Alexander Grothendieck introduced the concept in the early 1950s ( Grothendieck 1953 ), following the work of Dunford and Pettis, who developed earlier ...

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. 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. Approximation property - Wikipedia

   en.wikipedia.org/wiki/Approximation_property

   The construction of a Banach space without the approximation property earned Per Enflo a live goose in 1972, which had been promised by Stanisław Mazur (left) in 1936. [1] In mathematics, specifically functional analysis, a Banach space is said to have the approximation property (AP), if every compact operator is a limit of finite-rank ...

6. Anderson–Kadec theorem - Wikipedia

   en.wikipedia.org/wiki/Anderson–Kadec_theorem

   Eidelheit theorem: A Fréchet space is either isomorphic to a Banach space, or has a quotient space isomorphic to . Kadec renorming theorem: Every separable Banach space X {\displaystyle X} admits a Kadec norm with respect to a countable total subset A ⊆ X ∗ {\displaystyle A\subseteq X^{*}} of X ∗ . {\displaystyle X^{*}.}

7. 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.