Vol. 34, No. 2, 1970

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The strict topology on bounded sets

F. Dennis Sentilles

Vol. 34 (1970), No. 2, 529–540

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 →∥Axdetermines 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.

Mathematical Subject Classification
Primary: 46.50
Received: 7 November 1969
Published: 1 August 1970
F. Dennis Sentilles