Orlicz–Pettis theorem

OrliczPettis theorem

In functional analysis, the OrliczPettis theorem is a theorem about convergence in Banach spaces. It is named for Władysław Orlicz and Billy James Pettis. The result was originally proven by Orlicz for weakly sequentially complete normed spaces.[citation needed]

OrliczPettis theorem for normed spaces

Let X be a Banach space and let \left\{ {{x}_{n}} \right\} be any sequence in X. If the series \sum{{{x}_{n}}} is weakly subseries convergent, then the series is actually subseries convergent in the norm topology of X.