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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 ...
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.
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
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 ]
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 ...
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^{*}.}
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