Compact retractions and Schauder decompositions in Banach spaces

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21230%2F23%3A00370637" target="_blank" >RIV/68407700:21230/23:00370637 - isvavai.cz</a>
Result on the web
<a href="https://doi.org/10.1090/tran/8807" target="_blank" >https://doi.org/10.1090/tran/8807</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1090/tran/8807" target="_blank" >10.1090/tran/8807</a>

Result language
angličtina
Original language name
Compact retractions and Schauder decompositions in Banach spaces
Original language description
Let X be a separable Banach space. We give an almost characterization of the existence of a Finite Dimensional Decomposition (FDD for short) for X in terms of Lipschitz retractions onto generating compact subsets K of X. In one direction, if X admits an FDD then we construct a Lipschitz retraction onto a small generating convex and compact set K. On the other hand, we prove that if X admits a “small” generating compact Lipschitz retract then X has the π-property. It is still unknown if the π-property is isomorphically equivalent to the existence of an FDD. For dual Banach spaces this is true, so our results give a characterization of the FDD property for dual Banach spaces X. We give an example of a small generating convex compact set which is not a Lipschitz retract of C[0, 1], although it is contained in a small convex Lipschitz retract and contains another one. We characterize isomorphically Hilbertian spaces as those Banach spaces X for which every convex and compact subset is a Lipschitz retract of X. Finally, we prove that a convex and compact set K in any Banach space with a Uniformly Rotund in Every Direction norm is a uniform retract, of every bounded set containing it, via the nearest point map.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics

Project
<a href="/en/project/EF16_019%2F0000778" target="_blank" >EF16_019/0000778: Center for advanced applied science</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)<br>S - Specificky vyzkum na vysokych skolach

Publication year
2023
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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