Essays on the Orlicz-Pettis theorem. I: The two theorems (Q1175315)

There are discussed the Orlicz theorem form 1929 that a series \(\sum_ n x_ n\) of elements of a weakly sequentially complete Banach space \(X\) is unconditionally convergent in \(X\) if and only if \(\sum_ n| x^*(x_ n)|<\infty\) for every \(x^*\in X^*\), its version in Banach's book 1932, Pettis' formulation 1938 and their connections, applying vector measure approach.