Continuity of bilinear maps on direct sums of topological vector spaces

TL;DR

This paper establishes a criterion for the continuity of bilinear maps on countable direct sums of topological vector spaces, with applications to convolution maps and tensor algebras, solving open problems in the field.

Contribution

It introduces a new criterion for continuity of bilinear maps on direct sums and applies it to convolution and tensor algebra topologies, addressing open questions.

Findings

Convolution map on R^n is continuous.

Tensor algebra T(E) is a topological algebra iff sequences of seminorms on E are bounded.

T(E) is a topological algebra for metrizable E iff E is normable.

Abstract

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.