Continuity of bilinear maps on direct sums of topological vector spaces

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 f :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∈N 0 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 k ω . Classification: 46M05 (Primary); 42A85, 44A35, 46A13, 46A11, 46A16, 46E25, 46F05

Continuity of Convolution of Test Functions on Lie Groups

“…is discontinuous (as condition (b) from Theorem A is violated here). This had not been recorded yet in the works [30] and [22] devoted to G = R n .…”

Stochastic integration with respect to cylindrical semimartingales

“…First, observe that because the canonical inclusion S 0 → (S 0 ) lcx is linear and continuous, it follows from Theorem 3.3 that the stochastic integral mapping I induces a continuous linear operator from Φ ⊗ ν bP into (S 0 ) lcx . Then, since (Φ ⊗ ν bP) lcx ≃ Φ ⊗ π bP (see Lemma 6.3 in [12]) and (S 0 ) lcx is locally convex, it follows that (see [30], Proposition 6)…”

Stochastic integration with respect to cylindrical semimartingales

“…First, observe that because the canonical inclusion S 0 → (S 0 ) lcx is linear and continuous, it follows from Theorem 3.3 that the stochastic integral mapping I induces a continuous linear operator from Φ ⊗ ν bP into (S 0 ) lcx . Then, since (Φ ⊗ ν bP) lcx Φ ⊗ π bP (see Lemma 6.3 in [17]) and (S 0 ) lcx is locally convex, it follows that (see [42], Proposition 6)…”