## General Orlicz–Pettis theorem.(English)Zbl 1105.46008

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.

### MSC:

 46A99 Topological linear spaces and related structures

### Keywords:

$$G$$ valued duality pair; Orlicz-Pettis theorem
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### References:

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