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.^[1][2] The result was originally proven by Orlicz for weakly sequentially complete normed spaces.^[3]

Orlicz–Pettis Theorem for Banach 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.

References

1. Orlicz, W. (1929), "Beiträge zur Theorie der Orthogonalentwicklungen. I, II", Studia (in German) 1: 1–39, 241–255, Zbl 55.0164.02.
2. Pettis, B. J. (1938), "On integration in vector spaces", Transactions of the American Mathematical Society 44 (2): 277–304, doi:10.2307/1989973, MR 1501970.
3. Diestel, J.; Uhl, J. J., Jr. (1977), Vector Measures, Mathematical Surveys 15, Providence, R.I.: American Mathematical Society, p. 34, MR 0453964.