© Hindawi Publishing Corp. ON WEIGHTED SPACES WITHOUT A FUNDAMENTAL SEQUENCE OF BOUNDED SETS

The problem of countably quasi-barrelledness of weighted spaces of continuous functions, of which there are no results in the general setting of weighted spaces, is tackled in this paper. This leads to the study of quasi-barrelledness of weighted spaces in which, unlike that of Ernst and Schnettler (1986), though with a similar approach, we drop the assump-tion that the weighted space has a fundamental sequence of bounded sets. The study of countably quasi-barrelledness of weighted spaces naturally leads to definite results on the weighted (DF)-spaces for those weighted spaces with a fundamental sequence of bounded sets. 2000 Mathematics Subject Classification: 46A08, 46E30. 1. Introduction and notation