561 research outputs found
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
Quotients of Banach spaces and surjectively universal spaces
We characterize those classes of separable Banach spaces for
which there exists a separable Banach space not containing and
such that every space in the class is a quotient of .Comment: 23 pages, no figure
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
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