ABSTRACT
Throughout this paper the word ‘space’ will stand for ‘Hausdorff topological vector space over the field of real or complex numbers’. Let us recall that a space E is ultrabarrelled [16] ( £ -barrelled in [1]) if any linear mapping between E and any metrizable complete space with closed graph is continuous; E is £-suprabarrelled [23] if, given an increasing sequence of subspaces of E covering £ , there is one that is £ -barrelled and dense in E ; and E is totally £ -barrelled [23] if, given a sequence of subspaces of E covering E , there is one that is £ -barrelled and its closure is finite-codimensional in E . A countable number of ultrabarrelledness conditions was studied in [20] by calling £ -barrelled of class 1 to the £-suprabarrelled spaces and, for each positive integer n > 2, £ -barrelled of class n to any space E such that, given an increasing sequence of subspaces of E covering E , there is one that is £ -barrelled of class n — 1. Then a space E is said to be £ -barrelled of class No [20] if E is £ -barrelled of class n for each n e N.
