1,534 research outputs found
Simultaneously continuous retraction and Bishop-Phelps-Bollob\'as type theorem
We study the existence of a retraction from the dual space of a (real
or complex) Banach space onto its unit ball which is uniformly
continuous in norm topology and continuous in weak- topology. Such a
retraction is called a uniformly simultaneously continuous retraction.
It is shown that if has a normalized unconditional Schauder basis with
unconditional basis constant 1 and is uniformly monotone, then a
uniformly simultaneously continuous retraction from onto
exists. It is also shown that if is a family of separable Banach
spaces whose duals are uniformly convex with moduli of convexity
such that and or
for , then a uniformly simultaneously continuous retraction
exists from onto .
The relation between the existence of a uniformly simultaneously continuous
retraction and the Bishsop-Phelps-Bollob\'as property for operators is
investigated and it is proved that the existence of a uniformly simultaneously
continuous retraction from onto its unit ball implies that a pair has the Bishop-Phelps-Bollob\'as property for every locally compact
Hausdorff spaces . As a corollary, we prove that has the
Bishop-Phelps-Bollob\'as property if and are the spaces of
all real-valued continuous functions vanishing at infinity on locally compact
metric space and locally compact Hausdorff space respectively.Comment: 15 page
On continuous choice of retractions onto nonconvex subsets
For a Banach space and for a class \A of its bounded closed retracts,
endowed with the Hausdorff metric, we prove that retractions on elements A \in
\A can be chosen to depend continuously on , whenever nonconvexity of each
A \in \A is less than \f{1}{2}. The key geometric argument is that the set
of all uniform retractions onto an \a-paraconvex set (in the spirit of E.
Michael) is \frac{\a}{1-\a}-paraconvex subset in the space of continuous
mappings of into itself. For a Hilbert space the estimate
\frac{\a}{1-\a} can be improved to \frac{\a (1+\a^{2})}{1-\a^{2}} and the
constant \f{1}{2} can be reduced to the root of the equation \a+
\a^{2}+a^{3}=1
Polyfolds: A First and Second Look
Polyfold theory was developed by Hofer-Wysocki-Zehnder by finding
commonalities in the analytic framework for a variety of geometric elliptic
PDEs, in particular moduli spaces of pseudoholomorphic curves. It aims to
systematically address the common difficulties of compactification and
transversality with a new notion of smoothness on Banach spaces, new local
models for differential geometry, and a nonlinear Fredholm theory in the new
context. We shine meta-mathematical light on the bigger picture and core ideas
of this theory. In addition, we compiled and condensed the core definitions and
theorems of polyfold theory into a streamlined exposition, and outline their
application at the example of Morse theory.Comment: 62 pages, 2 figures. Example 2.1.3 has been modified. Final version,
to appear in the EMS Surv. Math. Sc
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