The Hardy spaces , the Sobolev spaces are examples of Banach spaces that are related to L p spaces and have additional structure. They are important in different branches of analysis, Harmonic analysis and Partial differential equations among others. Using the isometric embedding F X , it is customary to consider a normed space X as a subset of its bidual. In infinite-dimensional spaces, not all linear maps are continuous.
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Banach space - Wikipedia
Precisely, for every Banach space Xthe map. On every non-reflexive Epsace space Xthere exist continuous linear functionals that are not norm-attaining. They are important in different branches of analysis, Harmonic analysis and Partial differential equations among others. Find out more about the Kindle Personal Document Service.
For example, every convex continuous function on the unit ball B of a reflexive space attains its minimum at some point in B. When X has the approximation propertythis closure coincides with the space of compact operators on X. A normed space X is a Banach space if and only if each absolutely convergent series in X converges, .
If Z is another Banach space such banaxh there is an isometric isomorphism from X onto a dense subset of Zthen Z is isometrically isomorphic to Y.
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In other areas of analysisthe spaces under study are banaxh Banach spaces. Characterizing Hilbert Space Topology. This Banach space Y is the completion of the normed space X. This page was last edited on 11 Novemberat If X is infinite-dimensional, there exist linear maps which are not continuous.
Furthermore, just as Enflo's example, this space X is a "hand-made" space that fails to have the approximation property. Theory 2— There exists a canonical factorization of T as esapce.
List of Banach spaces. Further, by the open mapping theorem, if there is a bounded linear operator from the Banach space X onto the Banach space Ythen Y is reflexive.
 Espaces de Banach-Colmez et faisceaux coh\'erents sur la courbe de Fargues-Fontaine
If X and Y are normed spaces over the same ground field Kthe set of all continuous K -linear maps T: This was disproved by Gilles Pisier in Thus, the vector space B XY can be given the operator norm. Theory 3866 — This applies in particular to separable reflexive Banach spaces. Weak compactness of the unit ball provides a tool for finding solutions in reflexive spaces to certain optimization problems.
The normed space X is called reflexive when the natural map. If one of the two spaces X or Y is complete or reflexiveseparableetc.
This is a consequence of the Hahn—Banach theorem. Every normed space X can be isometrically baanch in a Banach space. The Schauder system is a basis in the space C [0, 1].
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Del and BenitezC. Conversely, when K 1 is not homeomorphic to K expacethe multiplicative Banach—Mazur distance between C K 1 and C K 2 must be greater than or equal to 2see above the results by Amir and Cambern.
Kadec's theorem was extended by Torunczyk, who proved  that any two Banach spaces are homeomorphic if and only if they have the same density characterthe minimum cardinality of a dense subset. A necessary and espave condition for the norm of a Banach space X to be associated to an inner product is the parallelogram identity:.
According to the Banach—Mazur theoremevery Banach space is isometrically isomorphic to a subspace of some C K. France 67—69 — espave, 23 —