On the German Wikipedia page on the Banach-Saks property, they claim that every Banach space with the Banach-Saks property is reflexive but that the converse is not true. There should be a proof due to T. Nishiura and D. Waterman, but unfortunately, I can't find a proof of this interesting statement. It would be appreciated if someone could give me an online available reference (if possible). Unfortunately, my library at the university doesn't have a copy.
Thx for your help
math
$\endgroup$ 12 Answers
$\begingroup$For the record: A Banach space $X$ has the Banach-Saks property if given a bounded sequence $(x_n)$ in $X$ there is a subsequence $(y_n)$ of $(x_n)$ such that the sequence $(\sigma_n)=(n^{-1}\sum\limits_{k=1}^n y_k)$ is norm convergent.
In Diestel's Sequences and Series in Banach spaces, the following outline for the proof that a Banach space with the Banach-Saks property is reflexive is given:
1) The Banach-Saks property is an isomorphic invariant.
2) If $X$ has the Banach-Saks property, then so do all of its closed linear subspaces.
3) $\ell_1$ does not have the Banach-Saks property.
4) If $(x_n)$ is weakly Cauchy and if ${\rm norm }\,\,\lim\limits_{n\rightarrow\infty} n^{-1} \sum\limits_{i=1}^n x_n$ exists, then $(x_n)$ is weakly convergent.
5) Conclude that a space with the Banach Saks property is reflexive.
Note by Rosenthal's $\ell_1$ theorem (Every bounded sequence in the Banach space $X$ has a weakly Cauchy subsequence if and only if $X$ contains no isomorphic copy of $\ell_1$), if $X$ has the Banach-Saks property, then by 1), 2), and 3), every bounded sequence has a weakly Cauchy subsequence. From 4), then, it follows that every bounded sequence in $X$ has a weakly convergent subsequence. Thus, the unit ball of $X$ is weakly compact by the Eberlein-Smulian Theorem; and so $X$ is reflexive.
$\endgroup$ 1 $\begingroup$Here is Nishiura and Waterman's paper.
Nishiura, T.; Waterman, D., Reflexivity and summability, Stud. Math. 23, 53-57 (1963). ZBL0121.09402.
Albert Baernstein gives a counter-example for the converse in the paper:
$\endgroup$ 1Baernstein, Albert II, On reflexivity and summability, Stud. Math. 42, 91-94 (1972). ZBL0228.46014.
