Search results
Results from the WOW.Com Content Network
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.
It is a closed linear subspace of the space of bounded sequences, , and contains as a closed subspace the Banach space of sequences converging to zero. The dual of c {\displaystyle c} is isometrically isomorphic to ℓ 1 , {\displaystyle \ell ^{1},} as is that of c 0 . {\displaystyle c_{0}.}
In the mathematical theory of Banach spaces, the closed range theorem gives necessary and sufficient conditions for a closed densely defined operator to have closed range. The theorem was proved by Stefan Banach in his 1932 Théorie des opérations linéaires .
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 ]
When is a Banach space, then this statement is true if and only if is a reflexive space. [2] Explicitly, a Banach space is reflexive if and only if for every closed proper vector subspace , there is some vector on the unit sphere of that is always at least a distance of = (,) away from the subspace.
Uniform Boundedness Principle — Let be a Banach space, a normed vector space and (,) the space of all continuous linear operators from into . Suppose that F {\displaystyle F} is a collection of continuous linear operators from X {\displaystyle X} to Y . {\displaystyle Y.}
Lomonosov's invariant subspace theorem is a mathematical theorem from functional analysis concerning the existence of invariant subspaces of a linear operator on some complex Banach space. The theorem was proved in 1973 by the Russian–American mathematician Victor Lomonosov .
If Σ is a sigma-algebra and μ is a sigma-additive positive measure on Σ then the Lp space L ∞ (μ) endowed with the essential supremum norm is by definition the quotient space of B(Σ) by the closed subspace of bounded μ-null functions: