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The precompactness-lemma. (English) Zbl 0599.46005
The purpose of this work is to draw attention to a result of Grothendieck which states that if $$<E,F>$$ is a dual system, $$A\subset E$$, $$B\subset F$$, then $$B^ 0$$-precompactness of A is equivalent to $$A^ 0$$- precompactness of B. A more general version due to Kakutani is also discussed briefly. Some elegant examples are given to demonstrate the applicability of this lemma.

##### MSC:
 46A20 Duality theory for topological vector spaces 46A50 Compactness in topological linear spaces; angelic spaces, etc. 46A32 Spaces of linear operators; topological tensor products; approximation properties 46G10 Vector-valued measures and integration