The precompactness-lemma. (English) Zbl 0599.46005

Floret, Klaus

Suppl. Rend. Circ. Mat. Palermo, II. Ser. 2, 75-82 (1982).

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

Keywords:

precompactness-lemma; Orlicz-Pettis-theorem on vector measures; Schwartz- spaces