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Theorem — If is a reflexive Banach space, every closed subspace of and every quotient space of are reflexive. This is a consequence of the Hahn–Banach theorem. Further, by the open mapping theorem, if there is a bounded linear operator from the Banach space X {\displaystyle X} onto the Banach space Y , {\displaystyle Y,} then Y ...
A closed subspace of ... James' space, a Banach space that has a Schauder basis, but has no unconditional Schauder Basis. Also, James' space is isometrically ...
All three spaces are complete (they are Banach spaces) with respect to the same norm defined by the total variation, and thus () is a closed subset of (), and () is a closed set of () for Σ the algebra of Borel sets on X.
The set of tempered distributions forms a vector subspace of the space of distributions ′ and is thus one example of a space of distributions; there are many other spaces of distributions. There also exist other major classes of test functions that are not subsets of C c ∞ ( U ) , {\displaystyle C_{c}^{\infty }(U),} such as spaces of ...
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.
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 .
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 .
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}.}