Basic sequences in non-Schwartz Fréchet spaces

Stability of Fréchet functional equation in non - Archimedean normed spaces ∗

We will establish stability of Fréchet functional equation

Nuclear spaces , Schwartz ’ kernel theorem

That is, maps from A to B∗ are given by integral kernels in (A⊗B)∗. However, the validity of this adjunction depends on existence of a genuine tensor product. We recall in an appendix the demonstration that infinitedimensional Hilbert spaces do not have tensor products. Also, we must specify the topology on the duals B∗ and (A⊗B)∗. The strongest conclusion gives these the strong topology, as co...

Bernstein's Lethargy Theorem in Fréchet spaces

In this paper we consider Bernstein’s Lethargy Theorem (BLT) in the context of Fréchet spaces. Let X be an infinite-dimensional Fréchet space and let V = {Vn} be a nested sequence of subspaces of X such that Vn ⊆ Vn+1 for any n ∈ N and X = ⋃∞ n=1 Vn. Let en be a decreasing sequence of positive numbers tending to 0. Under an additional natural condition on sup{dist(x, Vn)}, we prove that there e...

Basic Sequences in F-spaces and Their Applications

Unconditional basic sequences in spaces of large density

We study the problem of the existence of unconditional basic sequences in Banach spaces of high density. We show, in particular, the relative consistency with GCH of the statement that every Banach space of density אω contains an unconditional basic sequence.