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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.

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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.

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.