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

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). So...