Continuity of bilinear maps on direct sums of topological vector spaces

Details Continuity of bilinear maps on direct sums of topological vector spaces Continuity of bilinear maps on direct sums of topological vector spaces

2011-07-31

arxiv.org/abs/1108.0169v3

Mathematics

Functional Analysis

22 pages, LaTeX; v3: update of references

Scientific paper

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 taking a pair of test functions on R^n 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) as the locally convex direct sum of the projective tensor powers T^j(E) 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 whenever E is a DFS-space, or a hemicompact k-space.

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