We give an alternative proof of W. T. Gowers' theorem on block bases by reducing it to a discrete analogue on specific countable nets. We also give a Ramsey type result on k-tuples of block sequences in a normed linear space with a Schauder basis.
For a countable compact metric space and a seminormalized weakly null sequence (fₙ)ₙ in C() we provide some upper bounds for the norm of the vectors in the linear span of a subsequence of (fₙ)ₙ. These bounds depend on the complexity of and also on the sequence (fₙ)ₙ itself. Moreover, we introduce the class of c₀-hierarchies. We prove that for every α < ω₁, every normalized weakly null sequence (fₙ)ₙ in and every c₀-hierarchy generated by (fₙ)ₙ, there exists β ≤ α such that a sequence of β-blocks...
We introduce higher order spreading models associated to a Banach space X. Their definition is based on ℱ-sequences with ℱ a regular thin family and on plegma families. We show that the higher order spreading models of a Banach space X form an increasing transfinite hierarchy . Each contains all spreading models generated by ℱ-sequences with order of ℱ equal to ξ. We also study the fundamental properties of this hierarchy.
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