×

Density conditions on Fréchet and \((DF)\)-spaces. (English) Zbl 0745.46002

Summary: We survey our main results on the density condition for Fréchet spaces and on the dual density conditions for (DF)-spaces as well as some recent developments. At the end of section 1., we include a new result on the projective tensor product of two Fréchet spaces. Taskinen’s construction of counterexamples to Grothendieck’s “problème des topologies” yields Fréchet spaces \(E\), \(F\) with the density condition such that \(E\hat\otimes_ \pi F\) is not (even) distinguished. We prove now that the negative solution of the “problème des topologies” is, in fact, the only obstruction: For two Fréchet spaces \(E\) and \(F\) with the density condition, \(E\hat\otimes_ \pi F\) has the density condition as well (and hence is distinguished) whenever the “problème des topologies” has a positive solution for the pair \((E,F)\).

MSC:

46A04 Locally convex Fréchet spaces and (DF)-spaces
46M05 Tensor products in functional analysis
46A20 Duality theory for topological vector spaces
46A32 Spaces of linear operators; topological tensor products; approximation properties
46A45 Sequence spaces (including Köthe sequence spaces)
46A08 Barrelled spaces, bornological spaces
46M40 Inductive and projective limits in functional analysis
PDFBibTeX XMLCite
Full Text: EuDML