55 research outputs found
Tight embeddability of proper and stable metric spaces
We introduce the notions of almost Lipschitz embeddability and nearly
isometric embeddability. We prove that for , every proper
subset of is almost Lipschitzly embeddable into a Banach space if and
only if contains uniformly the 's. We also sharpen a result of N.
Kalton by showing that every stable metric space is nearly isometrically
embeddable in the class of reflexive Banach spaces.Comment: 19 page
Embeddings of locally finite metric spaces into Banach spaces
We show that if X is a Banach space without cotype, then every locally finite
metric space embeds metrically into X.Comment: 6 pages, to appear in Proceedings of the AM
Approximation properties and Schauder decompositions in Lipschitz-free spaces
We prove that the Lipschitz-free space over a doubling metric space has the
bounded approximation property. We also show that the Lipschitz-free spaces
over or have monotone finite-dimensional Schauder
decompositions
On Uniformly Convex and Uniformly Kadec-Klee Renomings
We give a new construction of uniformly convex norms with a power
type modulus on super-reflexive spaces based on the notion of dentability index.
Furthermore, we prove that if the Szlenk index of a Banach space is less than
or equal to Ï (first infinite ordinal) then there is an equivalent weak* lower
semicontinuous positively homogeneous functional on X* satisfying the uniform Kadec-Klee
Property for the weak*-topology (UKK*). Then we solve the UKK or UKK*
renorming problems for Lp(X) spaces and C(K) spaces for K scattered compact
space
Szlenk indices of convex hulls
We study the general measures of non-compactness defined on subsets of a dual
Banach space, their associated derivations and their -iterates. We
introduce the notions of convexifiable and sublinear measure of non-compactness
and investigate the properties of its associated fragment and slice
derivations. We apply our results to the Kuratowski measure of non-compactness
and to the study of the Szlenk index of a Banach space. As a consequence, we
obtain that the Szlenk index and the convex Szlenk index of a separable Banach
space are always equal. We also give, for any countable ordinal , a
characterization of the Banach spaces with Szlenk index bounded by
in terms of the existence of an equivalent renorming. This
extends a result by Knaust, Odell and Schlumprecht on Banach spaces with Szlenk
index equal to .Comment: This is the final revised version of this pape
Weak dentability index of spaces
We compute the weak-dentability index of the spaces where is a
countable compact space. Namely , whenever . More generally,
if is a scattered compact whose height
satisfies with an
countable
Approximation and Schur properties for Lipschitz free spaces over compact metric spaces
We prove that for any separable Banach space , there exists a compact
metric space which is homeomorphic to the Cantor space and whose Lipschitz-free
space contains a complemented subspace isomorphic to . As a consequence we
give an example of a compact metric space which is homeomorphic to the Cantor
space and whose Lipschitz-free space fails the approximation property and we
prove that there exists an uncountable family of topologically equivalent
distances on the Cantor space whose free spaces are pairwise non isomorphic. We
also prove that the free space over a countable compact metric space has the
Schur property. These results answer questions by G. Godefroy.Comment: 9 page
Isometric embeddings of compact spaces into Banach spaces
We show the existence of a compact metric space such that whenever
embeds isometrically into a Banach space , then any separable Banach space
is linearly isometric to a subspace of . We also address the following
related question: if a Banach space contains an isometric copy of the unit
ball or of some special compact subset of a separable Banach space , does it
necessarily contain a subspace isometric to ? We answer positively this
question when is a polyhedral finite-dimensional space, or .Comment: 8 page
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