In functional analysis, the Orlicz–Pettis 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.
Let X be a Banach space and let be any sequence in X. If the series is weakly subseries convergent, then the series is actually subseries convergent in the norm topology of X.