@misc{10481/78355,
year = {2021},
month = {11},
url = {https://hdl.handle.net/10481/78355},
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 Lips-
chitz 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  -property. We note that it is still
unknown if the  -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Lipschitz mappings due to J. Bourgain, and of an unpublished comple-
mentability 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.},
organization = {CAAS	CZ.02.1.01/0.0/0.0/16-019/0000778},
organization = {Ministry of Science and Innovation, Spain (MICINN)},
organization = {Spanish Government	PGC2018-093794-B-I00},
organization = {MIU (Spain)	FPU19/04085	
SGS21/056/OHK3/1T/13},
publisher = {American Mathematical Society},
keywords = {Lipschitz retractions},
keywords = {Approximation properties},
title = {Compact retractions and Schauder decompositions in Banach spaces},
author = {Hájek, Petr and Medina Sabino, Rubén},
}