A generalized Orlicz-Pettis Theorem and applications

Abstract

Recently there have been several generalizations of the classical Orlicz-Pettis Theorem, For example, Thomas ([13]) has shown that in the function space C(S), for S a compact Hausdorffspace, a series ~ f , is subseries convergent with respect to the topology of pointwise convergence on S iff ~ f , is subseries convergent with respect to the norm topology and then that the classical Orlicz-Pettis Theorem is a special case of this result. Thomas has also established several similar results for the pointwise topology on some of the familiar function spaces. Similarly, Diestel and Faires ([-6], 1.2) have shown that i fX is a Banach space and the dual X' contains no isomorphic copy of 1 ~176 then a series in X' is subseries convergent with respect to the weak* topology iff it is subseries convergent with respect to the norm topology. Finally, if E is a locally convex space, Tweddle ([14]) has shown that there is a strongest locally convex topology r on E such that every weak subseries convergent series is r-subseries convergent. Dierolf ([5]) has also given a description of the topology z as a polar topology. In this paper we use a form of the Mikusinski Diagonal Theorem due to Antosik ([1]) to establish an analogue of Tweddle's Theorem for group-valued series. In later sections we use the general form of the Orlicz-Pettis Theorem to treat several of the classical Orlicz-Pettis Theorems and results related to the Orlicz-Pettis Theorem. In Section 2 we establish the classical Orlicz-Pettis Theorem and the Diestel-Faires Theorem. In Section 3 we establish an Orlicz-Pettis Theorem for comact operators due to Kal ton ([11]). In Section4 we treat the Nikodym Convergence Theorem for both group-valued and operator-valued measures. In Section5 we consider Thomas ' results concerning the topology of pointwise convergence on the function spaces C(S) and I p.