Without using the notion of convex, but strictly only absolutely convex, Barrelled and Bornological locally convex spaces over an arbitrary field, which has a valuation and is complete with the metric induced by the valuation, are being studied. As a continuation of a paper by the same author, it is proven that a barrelled space X is the strict inductive limit of an increasing sequence of subspaces whose union is X and if the sequence consists of bounded sets, X is a \((DF)\) -space. Bornological spaces also being studied. Two results analogous to barrelled spaces follow: a finite codimensional subspace of a bornological space remains bornological, and the same is true for quasibarrelled instead of bornological.