Resolutions of topological linear spaces and continuity of linear maps

The main result of the paper is the following: If an F-space X is covered by a family (E alpha: alpha is an element of N-N) of sets such that E alpha subset of E beta whenever alpha <= beta and f is a linear map from X to a topological linear space Y which is continuous on each of the sets E, then f is continuous. This provides a very strong negative answer to a problem posed recently by J. Kakol and M. Lopez 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. (c) 2007 Elsevier Inc. All rights reserved.