Chapter 2: Subseries Convergence

In this chapter we study versions of the Orlicz–Pettis Theorem for subseries convergent series in abstract triples and use these results to establish versions of the theorem in various settings. We also give applications of the results to various topics in analysis. A series ${\displaystyle {\sum }_{j=1}^{\infty }{x}_j}$ in a topological Abelian group (X, τ) is τ subseries convergent if for every subsequence {n_j} the subseries ${\displaystyle {\sum }_{j=1}^{\infty }{x}_{n_j}}$ is τ convergent in X. The classical version of the Orlicz–Pettis Theorem for normed spaces asserts that a series in a normed space which is subseries convergent in the weak topology of the space is subseries convergent in the norm topology ([Or],[Pe]). The theorem has important applications to many areas in the integration of vector valued functions and vector valued measures. In particular, Pettis used the theorem to establish the countable additivity of the Pettis integral which he defined. The theorem has been extended to locally convex spaces and many other situations including topological groups. See [DU],[K1],[FL] for a discussion of the history of the subject. We refer to any result which asserts that a series which is subseries convergent in some weak topology is subseries convergent in a stronger topology as an Orlicz–Pettis Theorem…