Convex unconditionality and summability of weakly null sequences. (English) Zbl 0942.46007

The authors investigate the behaviour of the subsequences of a weakly null sequence \((x_n)_{n{\in}N}\) of a Banach space \(X\) with respect to two fundamental properties. The convex unconditionality is studied in the first section of the paper. It is proved that every normalized weakly null sequence in a Banach space \(X\) has a convexly unconditional subsequence (Theorem 1.3). In the second section of the paper, a hierarchy of summability methods is introduced and by using the considered hierarchy the authors give a complete classification of the complexity of weakly null sequences (the main results are given in Theorem 2.4.1).


46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
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