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If E is a locally convex topological vector space, let P(E) be thepre-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 theg-neighbourhood property in the sense of E. Jorda, i.e., spaces in which allsets M of continuous seminorms of cardinality up to g have an upper bound inP(E). On the other hand, we study bilinear maps b from a product of locallyconvex spaces E_1 and E_2 to a locally convex space F, which admit "productestimates" in the sense that for all p_{i,j} in P(F), i,j=1,2,..., there existp_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) forall x in E_1, y in E_2. The relations between these concepts are explored, andexamples given. The main applications concern spaces C^r_c(M,E)$ ofvector-valued test functions on manifolds.

Upper bounds for continuous seminorms and special properties of bilinear maps