Subseries convergence in the space of functions of bounded \(\phi\)- variation (Q1087784)

An infinite series \(\sum x_ i\) in an F-space X is said to be ''perfectly bounded'' if the set \(\{\sum \epsilon_ ix_ i:\epsilon_ i=0\) or \(1\}\) is bounded. It is called ''perfectly convergent'' if every subseries \(\sum \epsilon_ ix_ i\) is convergent, i.e., if the given series is unconditionally convergent. In this paper, conditions are determined for a series of elements in the space of functions of bounded \(\phi\)-variation to have the property that every perfectly bounded series is necessarily perfectly convergent. For example, a classical result due to Orlicz shows that in a weakly sequentially complete Banach space, every weakly bounded series has this property.