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

Orlicz, W. (1929), "Beiträge zur Theorie der Orthogonalentwicklungen. I, II", Studia (in German) 1: 1–39, 241–255, Zbl 55.0164.02.
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.
Diestel, J.; Uhl, J. J., Jr. (1977), Vector Measures, Mathematical Surveys 15, Providence, R.I.: American Mathematical Society, p. 34, MR 0453964.

Mathematics Encyclopedia