2011/12/08 by Helge Glöckner, Helge Glockner, Glockner, Helge
Mathematics · #22E30 #42A85 #44A35 #46A03 #46A11 #46A13 #46A32 #46E25 #46F05 (Primary) 22D15 #46H05 #46M05 (Secondary) #Advanced Banach Space Theory #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #Functional Equations Stability Results #math.FA #msc:22D15 #msc:22E30 #msc:42A85 #msc:44A35 #msc:46A03 #msc:46A11 #msc:46A13 #msc:46A32 #msc:46E25 #msc:46F05 #msc:46H05 #msc:46M05
paper · pdf · doi:10.48550/arxiv.1112.1824
24 pages, LaTeX; v3: additional references, minor changes to more traditional terminology
openalex publication_date 2011/12/08 · arxiv created 2012/05/16 · arxiv updated 2012/05/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 E1 and E2 to a locally convex space F, which admit "product estimates" in the sense that for all pi,j in P(F), i,j=1,2,..., there exist pi in P(E1) and qj in P(E2) such that pi,j(b(x,y)) <= pi(x)qj(y) for all x in E1, y in E2. The relations between these concepts are explored, and examples given. The main applications concern spaces Crc(M,E) of vector-valued test functions on manifolds.