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 of sets such that Eα⊂Eβ 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 Eα, then f is continuous. This provides a very strong negative answer to a problem posed recently by J. Ka̧kol 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. Ka̧kol, 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.