ABSTRACT
LECH DREWNOWSKI Faculty of Mathematics and Computer Science, A. Mickiewicz University, Matejki 48/49, 60-769 Poznan, Poland
This article is an expanded version of the talk delivered at the conference. In its first part (Sections 1 through 3), we describe briefly some of Professor Wladyslaw Orlicz’s concepts and results related to the unconditional (or subseries) convergence in Banach as well as more general spaces. This part wasn’t intended to be, and isn’t, a full survey of his achievements in this area, and the information included about subsequent developments is in no case complete. (For a comprehensive account of Orlicz’s scientific work, see [33, Part I].) We focus our attention on those ideas and results that are of general character, and completely skip their motivations or applications to particular problems (for instance, in the theory of orthogonal expansions). Thus the topics chosen for the presentation are: the Orlicz-Pettis theorem; spaces with Property (O), that is, those where a series is subseries convergent whenever its set of finite sums is bounded; and a result that we like to call the Orlicz Lo -theorem. This selection of topics is motivated by their direct relevance to the material of the second part of this article (Sections 5 through 7). Here, a sample of recent results of Iwo Labuda and the author [13]—[17] is presented, and these include: Orlicz-Pettis type theorems for spaces of measurable functions and general topological Riesz spaces; results on Property (O) and copies of co or /qo in such spaces; and a couple of results concerning the so called Lacunary Convergence Property. Section 4, on Kalton’s Orlicz-Pettis theorems, is sort of an intermission between the two main parts of the article.
1 THE ORLICZ-PETTIS THEOREM
