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Limit points of arithmetic means of sequences in Banach spaces

Roman Lávička (2000)

Commentationes Mathematicae Universitatis Carolinae

We shall prove the following statements: Given a sequence { a n } n = 1 in a Banach space 𝐗 enjoying the weak Banach-Saks property, there is a subsequence (or a permutation) { b n } n = 1 of the sequence { a n } n = 1 such that lim n 1 n j = 1 n b j = a whenever a belongs to the closed convex hull of the set of weak limit points of { a n } n = 1 . In case 𝐗 has the Banach-Saks property and { a n } n = 1 is bounded the converse assertion holds too. A characterization of reflexive spaces in terms of limit points and cores of bounded sequences is also given. The motivation for the...

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