2020/05/11 by Andrew J Wrobel, Wrobel, Andrew J.
Computer Science · Economics, Econometrics and Finance · Mathematics · #46B99 #46E30 (Primary) 46A70 #91B50 (Secondary) #Advanced Banach Space Theory #Economic theories and models #FOS: Economics and business #FOS: Mathematics #Functional Analysis (math.FA) #Optimization and Variational Analysis #Theoretical Economics (econ.TH)
paper · pdf · doi:10.48550/arxiv.2005.05202
openalex publication_date 2020/05/11 · openalex created_date 2020/05/13 · openalex updated_date 2026/07/28
Known results are reviewed about the bounded and convex bounded variants, bT and cbT, of a topology T on a real Banach space. The focus is on the cases of T = w(P*, P) and of T = m(P*, P), which are the weak* and the Mackey topologies on a dual Banach space P*. The convex bounded Mackey topology, cbm(P*, P), is known to be identical to m(P*, P). As for bm(P*, P), it is conjectured to be strictly stronger than m(P*, P) or, equivalently, not to be a vector topology (except when P is reflexive). Some uses of the bounded Mackey and the bounded weak* topologies in economic theory and its applications are pointed to. Also reviewed are the bounded weak and the compact weak topologies, bw(Y, Y*) and kw(Y, Y*), on a general Banach space Y, as well as their convex variants (cbw and ckw).