Continuity of bilinear maps on direct sums of topological vector spaces

We prove a criterion for continuity of bilinear maps on countable direct sums of topological vector spaces. As a first application, we get a new proof for the fact (due to Hirai et al., 2001) that the map f : C∞c (Rn)× C∞c (Rn)→C∞c (Rn), (γ, η) → γ ∗ η taking a pair of test functions to their convolution is continuous. The criterion also allows an open problem by K.-H. Neeb to be solved: If E is a locally convex space, regard the tensor algebra T (E) := j∈N0 T j (E) as the locally convex direct sum of projective tensor powers of E. We show that T (E) is a topological algebra if and only if every sequence of continuous seminorms on E has an upper bound. In particular, if E is metrizable, then T (E) is a topological algebra if and only if E is normable. Also, T (E) is a topological algebra if E is DFS or kω. © 2011 Elsevier Inc. All rights reserved