Compact retractions and Schauder decompositions in Banach spaces

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

Compact retractions and Schauder decompositions in Banach spaces

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

Compact retractions and Schauder decompositions in Banach spaces

Popis výsledku anglicky

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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/EF16_019%2F0000778" target="_blank" >EF16_019/0000778: Centrum pokročilých aplikovaných přírodních věd</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)<br>S - Specificky vyzkum na vysokych skolach

Rok uplatnění

2023

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Transactions of the American Mathematical Society

ISSN

0002-9947

e-ISSN

1088-6850

Svazek periodika

376

Číslo periodika v rámci svazku

2

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

30

Strana od-do

1343-1372

Kód UT WoS článku

000884855000001

EID výsledku v databázi Scopus

2-s2.0-85146466617