The Kadec-Pełczyński theorem in Lp, 1 ≤ p < 2

Berkes, István and Tichy, R. (2016) The Kadec-Pełczyński theorem in Lp, 1 ≤ p < 2. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 144 (5). pp. 2053-2066. ISSN 0002-9939

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Abstract

By a classical result of Kadec and Pełczyński (1962), every nor- malized weakly null sequence in Lp, p > 2, contains a subsequence equivalent to the unit vector basis of ℓ2 or to the unit vector basis of ℓp. In this paper we investigate the case 1 ≤ p < 2 and show that a necessary and sufficient condition for the first alternative in the Kadec-Pełczyński theorem is that the limit random measure μ of the sequence satisfies ∫R x2 dμ(x) ∈ Lp/2. © 2015 American Mathematical Society.

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