347 research outputs found
A fixed point theorem for the infinite-dimensional simplex
We define the infinite dimensional simplex to be the closure of the convex
hull of the standard basis vectors in R^infinity, and prove that this space has
the 'fixed point property': any continuous function from the space into itself
has a fixed point. Our proof is constructive, in the sense that it can be used
to find an approximate fixed point; the proof relies on elementary analysis and
Sperner's lemma. The fixed point theorem is shown to imply Schauder's fixed
point theorem on infinite-dimensional compact convex subsets of normed spaces.Comment: 8 pages; related work at http://www.math.hmc.edu/~su/papers.htm
The Banach Spaces L(Infinity)(C(0)) And C(0)(L(Infinity)) Are Not Isomorphic
The statement of the title is proved. It follows from this that the spaces c(0)(l(p)), l(p)(c(0)) and l(p)(l(q)), 1 <= p, q <= +infinity, make a family of mutually non-isomorphic Banach spaces
Isometry groups among topological groups
It is shown that a topological group G is topologically isomorphic to the
isometry group of a (complete) metric space iff G coincides with its
G-delta-closure in the Rajkov completion of G (resp. if G is Rajkov-complete).
It is also shown that for every Polish (resp. compact Polish; locally compact
Polish) group G there is a complete (resp. proper) metric d on X inducing the
topology of X such that G is isomorphic to Iso(X,d) where X = l_2 (resp. X = Q;
X = Q\{point} where Q is the Hilbert cube). It is demonstrated that there are a
separable Banach space E and a nonzero vector e in E such that G is isomorphic
to the group of all (linear) isometries of E which leave the point e fixed.
Similar results are proved for an arbitrary complete topological group.Comment: 30 page
Weak compactness of operators acting on o–O type spaces
We consider operators T : M_0 -> Z and T : M -> Z, where Z is a Banach space
and (M_0, M) is a pair of Banach spaces belonging to a general construction in
which M is defined by a "big-O" condition and M_0 is given by the corresponding
"little-o" condition. Prototype examples of such spaces M are given by
, weighted spaces of functions or their derivatives, bounded mean
oscillation, Lipschitz-H\"older spaces, and many others. The main result
characterizes the weakly compact operators T in terms of a certain norm
naturally attached to M, weaker than the M-norm, and shows that weakly compact
operators T : M_0 -> Z are already quite close to being completely continuous.
Further, we develop a method to extract c_0-subsequences from sequences in M_0.
Applications are given to the characterizations of the weakly compact
composition and Volterra-type integral operators on weighted spaces of analytic
functions, BMOA, VMOA, and the Bloch space.Comment: 12 page
Direct sums and the Szlenk index
For an ordinal and , we determine a necessary and
sufficient condition for an -direct sum of operators to have Szlenk
index not exceeding . It follows from our results that the
Szlenk index of an -direct sum of operators is determined in a natural
way by the behaviour of the -Szlenk indices of its summands. Our
methods give similar results for -direct sums.Comment: The proof of Proposition~2.4 has changed, with some of the arguments
transferred to the proof of an added-in lemma, Lemma~2.8. Changes have been
made to the Applications sectio
A topological characterization of LF-spaces
We present a topological characterization of LF-spaces and detect small
box-products that are (locally) homeomorphic to LF-spaces.Comment: 16 page
Functor of continuation in Hilbert cube and Hilbert space
A -set in a metric space is a closed subset of such that each
map of the Hilbert cube into can uniformly be approximated by maps of
into . The aim of the paper is to show that there exists a
functor of extension of maps between -sets of [or ] to maps acting
on the whole space [resp. ]. Special properties of the functor are
proved.Comment: 9 page
An introduction to nuclear space
This is a small book (48 pages) that contains a revised and extended version of the notes of seminar lectures given by Bessaga. The authors present a nice introduction to nuclear spaces (with all necessary preliminaries) based on Kolmogorov diameters. They consider only some of the most important topics of the theory of nuclear spaces, namely Kolmogorov diameters, nuclear operators, Mityagin's characterization of nuclear spaces, the theorem on absoluteness of bases in nuclear spaces, the uniqueness problem for bases (together with the theorem on quasiequivalence of regular bases), examples of nuclear Fréchet spaces without basis. Of course, many of the important topics in the theory of nuclear spaces are not even touched. Nevertheless, this book may be recommended to anyone who wants to study nuclear spaces, since it is practically independent of other sources and covers an essential part of the theory. Moreover, the authors do their best to help the reader: the proofs of all theorems are complete and the organization of the material is perfect
The absolutely continuous spectrum of one-dimensional Schr"odinger operators
This paper deals with general structural properties of one-dimensional
Schr"odinger operators with some absolutely continuous spectrum. The basic
result says that the omega limit points of the potential under the shift map
are reflectionless on the support of the absolutely continuous part of the
spectral measure. This implies an Oracle Theorem for such potentials and
Denisov-Rakhmanov type theorems.
In the discrete case, for Jacobi operators, these issues were discussed in my
recent paper [19]. The treatment of the continuous case in the present paper
depends on the same basic ideas.Comment: references added; a few very minor change
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