A class of finite dimensional decompositions (FDDs), called locally round, is introduced in Frechet spaces. A Frechet space with a locally round FDD can be viewed as a generalization of a Kothe space. The block subspaces and block quotients of such a space are always complemented and have a basis. Conversely, sometimes these properties characterize an FDD being locally round.

Këte spaces differing in diametral dimensionality

Nuclear Fréchet spaces with locally round finite dimensional decompositions

The classic diametral dimension is a topological invariant which characterizes Schwartz and nuclear locally convex spaces. Besides, there exists a second diametral dimension which is conjectured to be equal to the rst one (on Fréchet-Schwartz spaces). The rst part of this thesis is dedicated to the study of this conjecture. We present several positive partial results in metrizable spaces (in particular in Köthe sequence spaces and Hilbertizable spaces) and some properties which provide the equality of the two diametral dimensions (such as the ∆-stability, the existence of prominent bounded sets, and the property ( Ω ) ). Then, we describe the construction of some non-metrizable locally convex spaces for which the two diametral dimensions are di erent. The other purpose of this work is to pursue the topological study of sequence spaces Sν , originally de ned in the context of multifractal analysis. For this, the second part of the present thesis focuses on the study of the two diametral dimensions in spaces Sν . Finally, we show that some classes of spaces Sν verify (a variation of) the property ( Ω ) .

background

Diametral dimensions and some applications to spaces Snu

Espaces de suites, dimension diamétrale, propriétés (DN) et (Omega)

Isomorphisms of the so-called Koethe power spaces of the second kind are considered. These spaces are determined by a pair of sequences of positive numbers. Counting functions for a pair of sequences (m-rectangular characteristics of the corresponding Koethe space of the second kind) are introduced. They are shown to be invariant under isomorphisms. The proof is based on the construction of special compound invariants suitable for the class of spaces under consideration. New results on the linear topological structure of spaces of analytic functions in multicircular domains are obtained as an application. Bibliography: 29 titles.

Këte spaces differing in diametral dimensionality

Citations

Nuclear Fréchet spaces with locally round finite dimensional decompositions

Diametral dimensions and some applications to spaces Snu

Espaces de suites, dimension diamétrale, propriétés (DN) et (Omega)

Multirectangular characteristics for Köthe power spaces of the second kind

On stableD 1 andD 2 spaces

References

Approximative dimension of linear topological spaces and some of its applications

Related Papers (1)

Quasinormability and diametral dimension