# Orlicz–Pettis theorem

Orlicz–Pettis theorem A theorem in functional analysis concerning convergent series (Orlicz) or, equivalently, countable additivity of measures (Pettis) with values in abstract spaces.

Let {displaystyle X} be a Hausdorff locally convex topological vector space with dual {displaystyle X^{*}} . A series {displaystyle sum _{n=1}^{infty }~x_{n}} is subseries convergent (in {displaystyle X} ), if all its subseries {displaystyle sum _{k=1}^{infty }~x_{n_{k}}} are convergent. The theorem says that, equivalently, (i) If a series {displaystyle sum _{n=1}^{infty }~x_{n}} is weakly subseries convergent in {displaystyle X} (i.e., is subseries convergent in {displaystyle X} with respect to its weak topology {displaystyle sigma (X,X^{*})} ), then it is (subseries) convergent; or (ii) Let {displaystyle mathbf {A} } be a {displaystyle sigma } -algebra of sets and let {displaystyle mu :mathbf {A} to X} be an additive set function. If {displaystyle mu } is weakly countably additive, then it is countably additive (in the original topology of the space {displaystyle X} ).

The history of the origins of the theorem is somewhat complicated. In numerous papers and books there are misquotations or/and misconceptions concerning the result. Assuming that {displaystyle X} is weakly sequentially complete Banach space, W. Orlicz[1] proved the following Theorem. If a series {displaystyle sum _{n=1}^{infty }~x_{n}} is weakly unconditionally Cauchy, i.e., {displaystyle sum _{n=1}^{infty }|x^{*}(x_{n})|

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