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 : E 1 × E 2 → F between locally convex spaces which admit “product estimates” in the sense that for all p i , j ∈ P ( F ) , i , j = 1 , 2 , …  , there exist p i ∈ P ( E 1 ) and q j ∈ P ( E 2 ) such that p i , j ( β ( x , y ) ) ⩽ p i ( x ) q j ( y ) for all ( x , y ) ∈ E 1 × E 2 . The relations between these concepts are explored, and examples given. The main applications concern spaces C c r ( M , E ) of vector-valued test functions on manifolds.