Bounded topologies on Banach spaces and some of their uses in economic theory: a review

TL;DR

This review paper discusses various bounded and convex bounded topologies on Banach spaces, focusing on their properties, conjectures, and applications in economic theory, especially the Mackey and weak* topologies.

Contribution

It provides a comprehensive review of bounded topologies on Banach spaces, highlighting open conjectures and their relevance to economic applications.

Findings

cbm(P*, P) equals m(P*, P)

bm(P*, P) is conjectured to be strictly stronger than m(P*, P)

bounded Mackey and weak* topologies are useful in economic theory

Abstract

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).