  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).
