Upper bounds for continuous seminorms and special properties of bilinear maps

TL;DR

This paper investigates bounds for continuous seminorms in locally convex spaces and examines bilinear maps with product estimates, with applications to vector-valued test functions on manifolds.

Contribution

It introduces the g-neighbourhood property for locally convex spaces and analyzes bilinear maps with product estimates, connecting these concepts and providing relevant examples.

Findings

Spaces with the g-neighbourhood property are characterized.

Bilinear maps with product estimates are studied in detail.

Applications to spaces of vector-valued test functions are demonstrated.

Abstract

If E is a locally convex topological vector space, let P(E) be the pre-ordered set of all continuous seminorms on E. We study, on the one hand, for g an infinite cardinal those locally convex spaces E which have the g-neighbourhood property in the sense of E. Jorda, i.e., spaces in which all sets M of continuous seminorms of cardinality up to g have an upper bound in P(E). On the other hand, we study bilinear maps b from a product of locally convex spaces E_1 and E_2 to a locally convex space F, which admit "product estimates" in the sense that for all p_{i,j} in P(F), i,j=1,2,..., there exist p_i in P(E_1) and q_j in P(E_2) such that p_{i,j}(b(x,y)) <= p_i(x)q_j(y) for all x in E_1, y in E_2. The relations between these concepts are explored, and examples given. The main applications concern spaces C^r_c(M,E)$ of vector-valued test functions on manifolds.