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 , (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∈N0Tj(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.