# Subseries convergence

If $( G , \tau )$ is a Hausdorff Abelian topological group, a series $\sum x _ { k }$ in $G$ is $\tau$-subseries convergent (respectively, unconditionally convergent) if for each subsequence $\{ x _ { n_k } \}$ (respectively, each permutation $\pi$) of $\{ x_k \}$, the subseries $\sum _ { k = 1 } ^ { \infty } x _ {{ n } _ { k }}$ (respectively, the rearrangement $\sum _ { k = 1 } ^ { \infty } x _ { \pi ( k )}$) is $\tau$-convergent in $G$. In one of the early papers in the history of functional analysis, W. Orlicz showed that if $X$ is a weakly sequentially complete Banach space, then a series in $X$ 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 $X$, attempts have been made to replace the weak topology of $X$ by a weaker topology, $\sigma ( X , Y )$, generated by a subspace $Y$ of the dual space of $X$ which separates the points of $X$. Perhaps the best result in this direction is the Diestel–Faires theorem, which states that if $X$ contains no subspace isomorphic to $\text{l} ^ { \infty }$, then a series in $X$ is $\sigma ( X , Y )$ subseries convergent if and only if the series is norm subseries convergent. If $X$ is the dual of a Banach space $Z$ and $Y = Z$, then the converse also holds (see [a2], for references and further results).