Resolutions of topological linear spaces and continuity of linear maps

Abstract

The main result of the paper is the following: If an F -space X is covered by a family (E α : α ∈ N N ) of sets such that E α ⊂ E β whenever α β, and f is a linear map from X to a topological linear space Y which is continuous on each of the sets E α , then f is continuous. This provides a very strong negative answer to a problem posed recently by J. Kakol and M. López Pellicer. A number of consequences of this result are given, some of which are quite curious. Also, inspired by a related question asked by J. Kakol, it is shown that if a linear map is continuous on each member of a sequence of compact sets, then it is also continuous on every compact convex set contained in the linear span of the sequence. The construction applied to prove this is then used to interpret a natural linear topology associated with the sequence as the inductive limit topology in the sense of Ph. Turpin, and thus derive its basic properties.

A revised closed graph theorem for quasi-Suslin spaces

“…Note also that the assumption on E to be metrizable and complete cannot be dropped even if A α are compact, see [6,Remark 4.4 (ii)]. R. Pol proved, see [7], using Mycielski's theorem about independent functions, that an analytic vector subspace of a separable F -space has codimension either finite or 2 ℵ0 .…”

“…Does there exist an F -space which is an algebraic direct sum of two non-closed subspaces admitting a complete resolution? See also [6,Problem 4.21].…”

“…We need only F to be a tvs with a relatively countably compact resolution (RCC resolution shortly), whereas E is assumed (only) to be a Baire tvs. Theorem 1 applies to get an analytic graph type result stating that a linear map from a Baire tvs into a tvs whose graph admits a RCC resolution is continuous, and will be used to extend and provide another (and shorter) proofs of some recent results obtained by Drewnowski [6] about the continuity of special linear maps between F -spaces.…”

“…Hence metrizable complete separable topological vector spaces (tvs) and their countable inductive limits admit compact resolutions [17,Theorem 2.1]. Very recently Drewnowski [6] has used the notion of resolution to prove some nice results about certain continuous maps between Fréchet and Banach spaces. Drewnowski's paper [6] motivated us to look again at Valdivia's classic closed graph theorem for quasi-Suslin spaces [18, I.4.2 (11)], which states that a linear mapping with closed graph from a metrizable Baire locally convex space (lcs) E into a quasi-Suslin lcs F is continuous.…”

“…Very recently Drewnowski [6] has used the notion of resolution to prove some nice results about certain continuous maps between Fréchet and Banach spaces. Drewnowski's paper [6] motivated us to look again at Valdivia's classic closed graph theorem for quasi-Suslin spaces [18, I.4.2 (11)], which states that a linear mapping with closed graph from a metrizable Baire locally convex space (lcs) E into a quasi-Suslin lcs F is continuous. So we use some techniques of [18] and of our own to get a closed graph theorem (Theorem 1) which extends Valdivia's [18, I.4.2 (11)] as well as Drewnowski's [6, Corollary 4.10, Corollary 4.11].…”