# Subseries convergence

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If is a Hausdorff Abelian topological group, a series in is -subseries convergent (respectively, unconditionally convergent) if for each subsequence (respectively, each permutation ) of , the subseries (respectively, the rearrangement ) is -convergent in . In one of the early papers in the history of functional analysis, W. Orlicz showed that if is a weakly sequentially complete Banach space, then a series in is weakly unconditionally convergent if and only if the series is norm unconditionally convergent [a5]. Later, he noted that if "unconditional convergence" is replaced by "subseries convergence" , the proof showed that the weak sequential completeness assumption could be dropped. That is, a series in a Banach space is weakly subseries convergent if and only if the series is norm subseries convergent; this result was announced in [a1], but no proof was given. In treating some problems in vector-valued measure and integration theory, B.J. Pettis needed to use this result but noted that no proof was supplied and then proceeded to give a proof ([a6]; the proof is very similar to that of Orlicz). The result subsequently came to be known as the Orlicz–Pettis theorem (see [a3] for a historical discussion).
In the case of a Banach space , attempts have been made to replace the weak topology of by a weaker topology, , generated by a subspace of the dual space of which separates the points of . Perhaps the best result in this direction is the Diestel–Faires theorem, which states that if contains no subspace isomorphic to , then a series in is subseries convergent if and only if the series is norm subseries convergent. If is the dual of a Banach space and , then the converse also holds (see [a2], for references and further results).