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A QUANTITATIVE EXTENSION OF SZLENK’S THEOREM
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 100 / Issue 2 / October 2019
- Published online by Cambridge University Press:
- 20 February 2019, pp. 338-343
- Print publication:
- October 2019
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- Article
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We show that for a bounded subset
$A$ of the 
$L_{1}(\unicode[STIX]{x1D707})$ space with finite measure 
$\unicode[STIX]{x1D707}$, the measure of weak noncompactness of 
$A$ based on the convex separation of sequences coincides with the measure of deviation from the Banach–Saks property expressed by the arithmetic separation of sequences. A similar result holds for a related quantity with the alternating signs Banach–Saks property. The results provide a geometric and quantitative extension of Szlenk’s theorem saying that every weakly convergent sequence in the Lebesgue space 
$L_{1}$ has a subsequence whose arithmetic means are norm convergent.
