We show by example that the associative law does not hold for tensor products in the category of general (not necessarily locally convex) topological vector spaces. The same pathology occurs for tensor products of Hausdorff abelian topological groups.
Answering a question of Halbeisen we prove (by two different methods) that the algebraic dimension of each infinite-dimensional complete linear metric space X equals the size of X. A topological method gives a bit more: the algebraic dimension of a linear metric space X equals |X| provided the hyperspace K(X) of compact subsets of X is a Baire space. Studying the interplay between Baire properties of a linear metric space X and its hyperspace, we construct a hereditarily Baire linear metric space...
We construct a quasi-Banach space X which contains no basic sequence.
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...
Let R be a subcategory of the category of all topological vector spaces. Let E be an element of R. The problem of the existence of the finest R-topology on E with the same continuous linear functionals as the original one is discussed. Remarks concerning the Hahn-Banach Extension Property are included.
Let X be a real or complex vector space equipped with the strongest vector space topology . Besides the result announced in the title we prove that X is uncountable-dimensional if and only if it is not locally pseudoconvex.
We examine the so-called three-space-stability for some classes of linear topological and locally convex spaces for which this problem has not been investigated.
We discuss various results on the existence of ‘true’ preimages under continuous open maps between -spaces, -lattices and some other spaces. The aim of the paper is to provide accessible proofs of this sort of results for functional-analysts.