Upper bounds for continuous seminorms and special properties of bilinear maps

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 θ an infinite cardinal those locally convex spaces E which have the θ-neighbourhood property introduced by E. Jordá, meaning that all sets M of continuous seminorms of cardinality |M|⩽θ have an upper bound in P(E). On the other hand, we study bilinear maps β:E1×E2→F between locally convex spaces which admit “product estimates” in the sense that for all pi,j∈P(F), i,j=1,2,… , there exist pi∈P(E1) and qj∈P(E2) such that pi,j(β(x,y))⩽pi(x)qj(y) for all (x,y)∈E1×E2. The relations between these concepts are explored, and examples given. The main applications concern spaces of vector-valued test functions on manifolds.