If B is a Banach algebra with
approximate identity and the Banach space X is a left B-module, the strict topology
,8 on X is the topology given by the seminorms x →∥Tx∥, one for each
T ∈ B. It is shown that β is the finest locally convex topology on X agreeing
with itself on the bounded sets in X, and that in certain circumstances a
single semi-norm x →∥Ax∥ determines β on each bounded set. It is then
natural to investigate the sufficiency of sequences in determining the strict
topology. A study is made of the finest locally convex topology on X having the
same convergent sequences as β, and sufficient conditions are given which
place the strict topology in the context of earlier sequential studies of other
authors.
