535 research outputs found
Operators between subspaces and quotients of L1
We provide an unified approach of results of L. Dor on the complementation of
the range, and of D. Alspach on the nearness from isometries, of small into
isomorphisms of L1. We introduce the notion of small subspace of L1 and show
lifting theorems for operators between quotients of L1 by small subspaces. We
construct a subspace of L1 which shows that extension of isometries from
subspaces of L1 to the whole space are no longer true for isomorphisms, and
that nearly isometric isomorphisms from subspaces of L1 into L1 need not be
near from any isometry.Comment: 35 page
A new metric invariant for Banach spaces
We show that if the Szlenk index of a Banach space is larger than the
first infinite ordinal or if the Szlenk index of its dual is larger
than , then the tree of all finite sequences of integers equipped with
the hyperbolic distance metrically embeds into . We show that the converse
is true when is assumed to be reflexive. As an application, we exhibit new
classes of Banach spaces that are stable under coarse-Lipschitz embeddings and
therefore under uniform homeomorphisms.Comment: 22 page
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