ε-Weak Cauchy sequences and a quantitative version of Rosenthal's l1-theorem

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:434

Article

Gasparis, I.¹ 

[1] Natl Tech Univ Athens, Sch Appl Math & Phys Sci, Athens 15780, Greece

关键词: The l(1)-theorem;    Weak Cauchy sequence;    Ramsey theorem;   

DOI  :  10.1016/j.jmaa.2015.09.079

来源: Elsevier

【 摘 要 】

A bounded sequence (x(n)) in a Banach space is called epsilon-weak Cauchy, for some epsilon > 0, if for all x* is an element of B-X. there exists some n(0) is an element of N such that vertical bar x*(x(n)) - x*(x(m))vertical bar < epsilon for all n >= n(0) and m >= n(0). It is shown that given epsilon > 0 and a bounded sequence (x(n)) in a Banach space then either (x(n)) admits an epsilon-weak Cauchy subsequence or, for all delta > 0, there exists a subsequence (x(mn)) with the following property. If I is a finite subset of N and phi: I -> N\I is any map then parallel to Sigma(n is an element of I) lambda(n) (x(mn) - x(m phi(n)))parallel to >= (epsilon/pi - delta) Sigma(n is an element of I) vertical bar lambda(n)vertical bar for every sequence of complex scalars (lambda(n))(n is an element of I) . This provides an alternative proof for Rosenthal's l(1)-theorem and strengthens its quantitative version due to Behrends. As a corollary we obtain that for any uniformly bounded sequence (f(n)) of complex valued functions, continuous on the compact Hausdorff space K and satisfying lim sup(n,m ->infinity) vertical bar f(n)(t) - f(m)(t)vertical bar <= epsilon, for some epsilon > 0 and all t is an element of K, there exists a subsequence (f(jn)) satisfying lim sup(n,m ->infinity) vertical bar integral(K)(f(jn) - f(jm))d mu vertical bar <= 2 epsilon, for every Radon measure mu on K. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

Free