1,190 research outputs found
The complexity of classifying separable Banach spaces up to isomorphism
It is proved that the relation of isomorphism between separable Banach spaces
is a complete analytic equivalence relation, i.e., that any analytic
equivalence relation Borel reduces to it. Thus, separable Banach spaces up to
isomorphism provide complete invariants for a great number of mathematical
structures up to their corresponding notion of isomorphism. The same is shown
to hold for (1) complete separable metric spaces up to uniform homeomorphism,
(2) separable Banach spaces up to Lipschitz isomorphism, and (3) up to
(complemented) biembeddability, (4) Polish groups up to topological
isomorphism, and (5) Schauder bases up to permutative equivalence. Some of the
constructions rely on methods recently developed by S. Argyros and P. Dodos
Bounded time computation on metric spaces and Banach spaces
We extend the framework by Kawamura and Cook for investigating computational
complexity for operators occurring in analysis. This model is based on
second-order complexity theory for functions on the Baire space, which is
lifted to metric spaces by means of representations. Time is measured in terms
of the length of the input encodings and the required output precision. We
propose the notions of a complete representation and of a regular
representation. We show that complete representations ensure that any
computable function has a time bound. Regular representations generalize
Kawamura and Cook's more restrictive notion of a second-order representation,
while still guaranteeing fast computability of the length of the encodings.
Applying these notions, we investigate the relationship between purely metric
properties of a metric space and the existence of a representation such that
the metric is computable within bounded time. We show that a bound on the
running time of the metric can be straightforwardly translated into size bounds
of compact subsets of the metric space. Conversely, for compact spaces and for
Banach spaces we construct a family of admissible, complete, regular
representations that allow for fast computation of the metric and provide short
encodings. Here it is necessary to trade the time bound off against the length
of encodings
Isomorphic Schauder decompositions in certain Banach spaces
We extend a theorem of Kato on similarity for sequences of projections in
Hilbert spaces to the case of isomorphic Schauder decompositions in certain
Banach spaces. To this end we use -Hilbertian and
-Hilbertian Schauder decompositions instead of orthogonal Schauder
decompositions, generalize the concept of an orthogonal Schauder decomposition
in a Hilbert space and introduce the class of spaces with Schauder-Orlicz
decompositions. Furthermore, we generalize the notions of type, cotype,
infratype and -cotype of a Banach space and study the properties of
unconditional Schauder decompositions in spaces possessing certain geometric
structure.Comment: 35 page
A strong boundedness result for separable Rosenthal compacta
It is proved that the class of separable Rosenthal compacta on the Cantor set
having a uniformly bounded dense sequence of continuous functions, is strongly
bounded.Comment: 13 pages, no figure
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