Every locally convex Hausdorff topological vector space has a base of neighbourhoods of the origin formed by barrels. A space E is barrelled if every barrel contains a neighbourhood of 0. For example, every Banach space is barrelled. A barrel B 'subset' E is a closed, convex, balanced, absorbing set. Suppose E is a locally convex space.
L implies that there are no measurable cardinals.
So it is consistent relative to ZFC that every bounded product of barrelled normed spaces is barrelled. The question is thus whether it is a theorem of ZFC that bounded products of barrelled normed spaces preserve barrelledness, or whether a large cardinal axiom will enable one to find a counterexample. There is also a characterisation of barrelled products in terms of ultrapowers. Let S be a set and E be a barrelled normed space.