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Discrete Copies of Rings of Sets in Groups and Orlicz-Pettis Theorems

Published online by Cambridge University Press:  20 November 2018

Iwo Labuda
Iwo Labuda*
Affiliation:
University of Florida, Gainesville, Florida; Mathematical Institute of The Polish Academy of Sciences, ul. Mielzynskiego 27/29, 61725 Poznan, Poland
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The Orlicz-Pettis type theorems are essentially statements about the equivalence of strong and weak subseries convergence of series in topological groups. The corresponding problem of the equivalence of subfamily summability for summable families was examined for instance in [2] and (16). We give here another uncountable generalization of the Orlicz-Pettis theorem in the sense that we assume less than subfamily summability and obtain straightforward generalizations of the classical case, which include or imply in a trivial manner—at least for polar topologies on groups—all known theorems “from weak to strong σ-additivity”.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 30 , Issue 4 , 01 August 1978 , pp. 748 - 755
Copyright
Copyright © Canadian Mathematical Society 1978

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