Compact retractions and Schauder decompositions in Banach spaces

TL;DR

This paper explores the relationship between finite dimensional decompositions (FDDs) in Banach spaces and Lipschitz retractions onto small convex compact sets, providing characterizations and examples relevant to Banach space geometry.

Contribution

It establishes a connection between FDDs and Lipschitz retractions, characterizes FDDs in dual spaces via Lipschitz retractions, and characterizes Hilbert spaces through Lipschitz retracts of convex compact sets.

Findings

FDD implies Lipschitz retraction onto small convex compact sets

Existence of Lipschitz retraction implies the $ extpi$-property in Banach spaces

Characterization of Hilbert spaces via Lipschitz retracts of convex compact sets

Abstract

In our note we show the very close connection between the existence of a Finite Dimensional Decomposition (FDD for short) for a separable Banach space $X$ and the existence of a Lipschitz retraction of $X$ onto a small (in a certain precise sense) generating convex and compact subset $K$ 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. We note that 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 lead in particular to a characterization of the FDD property for dual Banach spaces $X$ in terms of the existence of Lipschitz retractions onto small generating convex and compact subsets of $X$. It is conceivable that our results will find applications in the area of Lipschitz isomorphisms of Banach spaces. Our arguments make critical use of the Lipschitzization of coarse Lipschitz mappings due to J. Bourgain, and of an unpublished complementability result of V. Milman. 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. In the last part of our note 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.