Orlicz–Pettis theorem

Orlicz–Pettis theorem: 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.^{[citation needed]}

Orlicz–Pettis 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.

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