Compact retractions and Schauder decompositions in Banach spaces
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- by Petr Hájek and Rubén Medina;
- Trans. Amer. Math. Soc. 376 (2023), 1343-1372
- DOI: https://doi.org/10.1090/tran/8807
- Published electronically: November 9, 2022
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Abstract:
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 $\pi$-property. It is still unknown if the $\pi$-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.
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Bibliographic Information
- Petr Hájek
- Affiliation: Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague, Technická 2, 166 27 Praha 6, Czech Republic
- Email: hajek@math.cas.cz
- Rubén Medina
- Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain; and Department of Mathematics, Faculty of Electrical Engineering, Czech technical University in Prague, Technická 2, 166 27 Praha, Czech Republic
- ORCID: 0000-0002-4925-0057
- Email: rubenmedina@ugr.es
- Received by editor(s): December 6, 2021
- Received by editor(s) in revised form: June 18, 2022, and August 5, 2022
- Published electronically: November 9, 2022
- Additional Notes: This research was supported by CAAS CZ.02.1.01/0.0/0.0/16-019/0000778 and by the project SGS21/056/OHK3/1T/13. The second author’s research was also supported by MICINN (Spain) Project PGC2018-093794-B-I00 and MIU (Spain) FPU19/04085 Grant.
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 376 (2023), 1343-1372
- MSC (2020): Primary 46B20, 46B80, 54C55
- DOI: https://doi.org/10.1090/tran/8807
- MathSciNet review: 4531677