General Orlicz–Pettis theorem. (English) Zbl 1105.46008

Cui, Chengri; Wen, Songlong

Z. Anal. Anwend. 24, No. 3, 637-648 (2005).

Let \(G\) be a topological group and \(E\) a non-empty set. Let \(F\subset G^E\). Then \(E,F\) is called a \(G\) valued duality pair. Let \(w(E,F)\) (\(w(F,E)\)) be the weakest topology on \(E\) (\(F\)) such that all the maps \(x\mapsto f(x)\) (\(f\mapsto f(x)\)) for \(f\in F\) (\(x\in E\)) are continuous. A sequence \(\{x_j\}\) in \(E\) is \(w(E,F)\) subseries convergent if for every \( \sigma \subset\mathbb{N}\) there exists \(x_{\sigma }\in E\) such that \(\sum_{j\in \sigma }f(x_j)=f(x_{\sigma })\) for every \(f\in F\). Let \(\mathcal{F}\) be a family of subsets of \(F\). A sequence \(\{x_j\}\) in \(E\) is subseries convergent in the topology of uniform convergence on \(\mathcal{F}\) if for every \(B\in \mathcal{F}\) and for every \(\sigma \subset\mathbb{N}\) there exists \(x_{\sigma }\in E\) such that \(\sum_{j\in \sigma}f(x_{j})=f(x_{\sigma })\) uniformly for \(f\in B\). A subset \(B\subset F\) is conditionally \(w(F,E)\) sequentially compact if every sequence \(\{f_j\}\) has a subsequence \(\{f_{n_j}\}\) such that \(\lim f_{n_j}(x)\) exists for every \(x\in E\).
The authors prove a very abstract version of the Orlicz–Pettis theorem. They show that if \(\{x_j\}\) is \(w(E,F)\) subseries convergent, then \(\{x_j\}\) is subseries convergent in the topology of uniform convergence on conditionally \(w(F,E)\) sequentially compact (\(w(F,E)\) compact, \(w(F,E)\) countably compact) subsets of \(F\). Applications of the abstract Orlicz–Pettis theorem are given to the Nikodym convergence theorem, the Hahn–Schur theorem, and several other versions of the Orlicz–Pettis theorem.

Reviewer: Charles Swartz (Las Cruces)

MSC:

46A99

Topological linear spaces and related structures

Keywords:

\(G\) valued duality pair; Orlicz-Pettis theorem

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