## 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).

### MSC:

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

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