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It is proved that every normalized weakly null sequence has a subsequence which is convexly unconditional. Further, an Hierarchy of summability methods is introduced and with this we give a complete classification of the complexity of weakly null sequences.

We show that every Banach space X containing an isomorphic copy of c 0 has an infinite equilateral set and also that if X has a bounded biorthogonal system of size α then it can be renormed so as to admit an equilateral set of equal size.

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We show that for "most" compact non metrizable spaces, the unit ball of the Banach space C(K) contains an uncountable 2-equilateral set. We also give examples of compact non metrizable spaces K such that the minimum cardinality of a maximal equilateral set in C(K) is countable. γ∈Γ K γ of nontrivial compact metric 2010 Mathematics Subject Classification: Primary 46B20, 46E15;Secondary 46B26,54D30. Key words and phrases: equilateral set, maximal equilateral set, linked family.2. Let K be a compact totally disconnected space, then the family {(V, K \V ) :V is a clopen subset of K} is linked.