Compact retractions and Schauder decompositions in Banach spaces

Hájek, Petr; Medina, Rubén

doi:10.1090/tran/8807

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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