Tame Köthe Sequence Spaces are Quasi-Normable
We show that every tame Fréchet space admits a continuous norm and that every tame Köthe sequence space is quasi-normable.
Tame Spaces and Power Series Spaces.
Tensor Sequences and Inductive Limits with Local Partition of Unity.
Tensor stable Fréchet and (DF)-spaces.
In this paper we introduce and investigate classes of Fréchet and (DF)-spaces which constitute a very general frame in which the problem of topologies of Grothendieck and some related dual questions have a positive answer. Many examples of spaces in theses classes are provided, in particular spaces of sequences and functions. New counterexamples to the problems of Grothendieck are given.
The Ascoli property for function spaces and the weak topology of Banach and Fréchet spaces
Following Banakh and Gabriyelyan (2016) we say that a Tychonoff space X is an Ascoli space if every compact subset of is evenly continuous; this notion is closely related to the classical Ascoli theorem. Every -space, hence any k-space, is Ascoli. Let X be a metrizable space. We prove that the space is Ascoli iff is a -space iff X is locally compact. Moreover, endowed with the weak topology is Ascoli iff X is countable and discrete. Using some basic concepts from probability theory and...
The chi function in generalized summability
The concept of boundedness and the Bohr compactification of a MAP Abelian group
Let G be a maximally almost periodic (MAP) Abelian group and let ℬ be a boundedness on G in the sense of Vilenkin. We study the relations between ℬ and the Bohr topology of G for some well known groups with boundedness (G,ℬ). As an application, we prove that the Bohr topology of a topological group which is topologically isomorphic to the direct product of a locally convex space and an -group, contains “many” discrete C-embedded subsets which are C*-embedded in their Bohr compactification. This...
The density condition and the strong dual density condition by operator.
The aim of the present article is to introduce and investigate topological properties by operator. We obtain good stability properties for the density condition and the strong dual density condition by taking injective tensor products. Further we analyze the connection to (DF)-properties by operator.
The density condition in projective tensor products.
In this paper we modify a construction due to J. Taskinen to get a Fréchet space F which satisfies the density condition such that the complete injective tensor product l2 x~eF'b does not satisfy the strong dual density condition of Bierstedt and Bonet. In this way a question that remained open in Heinrichs (1997) is solved.
The density condition in quotients of quasinormable Fréchet spaces
It is proved that a separable Fréchet space is quasinormable if, and only if, every quotient space satisfies the density condition of Heinrich. This answers positively a conjecture of Bonet and Dí az in the class of separable Fréchet spaces.
The density condition in quotients of quasinormable Fréchet spaces, II.
It is proved that a Fréchet space is quasinormable if, and only if, every quotient space satisfies the density condition of Heinrich. This answers positively a conjecture of Bonet and Díaz
The Density Condition in Subspaces and Quotients of Frechet Spaces.
The exactness of the projective limit functor on the category of quotients of Frechet spaces
We give conditions under which the functor projective limit is exact on the category of quotients of Fréchet spaces of L. Waelbroeck [18].
The Fréchet space admits a strictly stronger separable and quasicomplete locally convex topology
The Hypercyclicity Criterion for sequences of operators
We show that under no hypotheses on the density of the ranges of the mappings involved, an almost-commuting sequence (Tₙ) of operators on an F-space X satisfies the Hypercyclicity Criterion if and only if it has a hereditarily hypercyclic subsequence , and if and only if the sequence (Tₙ ⊕ Tₙ) is hypercyclic on X × X. This strengthens and extends a recent result due to Bès and Peris. We also find a new characterization of the Hypercyclicity Criterion in terms of a condition introduced by Godefroy...
The Mackey completions of some interpolation F-spaces
The metric space , 0
The non-pluripolarity of compact sets in complex spaces and the property for the space of germs of holomorphic functions
The aim of this paper is to establish the equivalence between the non-pluripolarity of a compact set in a complex space and the property for the dual space of the space of germs of holomorphic functions on that compact set.
The Open Mapping and Closed Range Theorems.
The projective tensor product of Fréchet-Montel spaces