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Characterizations of elements of a double dual Banach space and their canonical reproductions

Vassiliki Farmaki — 1993

Studia Mathematica

For every element x** in the double dual of a separable Banach space X there exists the sequence ( x ( 2 n ) ) of the canonical reproductions of x** in the even-order duals of X. In this paper we prove that every such sequence defines a spreading model for X. Using this result we characterize the elements of X**╲ X which belong to the class B 1 ( X ) B 1 / 2 ( X ) (resp. to the class B 1 / 4 ( X ) ) as the elements with the sequence ( x ( 2 n ) ) equivalent to the usual basis of 1 (resp. as the elements with the sequence ( x ( 4 n - 2 ) - x ( 4 n ) ) equivalent to the usual basis...

On spreading c 0 -sequences in Banach spaces

Vassiliki Farmaki — 1999

Studia Mathematica

We introduce and study the spreading-(s) and the spreading-(u) property of a Banach space and their relations. A space has the spreading-(s) property if every normalized weakly null sequence has a subsequence with a spreading model equivalent to the usual basis of c 0 ; while it has the spreading-(u) property if every weak Cauchy and non-weakly convergent sequence has a convex block subsequence with a spreading model equivalent to the summing basis of c 0 . The main results proved are the following: (a)...

Ordinal indices and Ramsey dichotomies measuring c₀-content and semibounded completeness

Vassiliki Farmaki — 2002

Fundamenta Mathematicae

We study the c₀-content of a seminormalized basic sequence (χₙ) in a Banach space, by the use of ordinal indices (taking values up to ω₁) that determine dichotomies at every ordinal stage, based on the Ramsey-type principle for every countable ordinal, obtained earlier by the author. We introduce two such indices, the c₀-index ξ ( χ ) and the semibounded completeness index ξ b ( χ ) , and we examine their relationship. The countable ordinal values that these indices can take are always of the form ω ζ . These results...

Extended Ramsey theory for words representing rationals

Vassiliki FarmakiAndreas Koutsogiannis — 2013

Fundamenta Mathematicae

Ramsey theory for words over a finite alphabet was unified in the work of Carlson, who also presented a method to extend the theory to words over an infinite alphabet, but subject to a fixed dominating principle. In the present work we establish an extension of Carlson's approach to countable ordinals and Schreier-type families developing an extended Ramsey theory for dominated words over a doubly infinite alphabet (in fact for ω-ℤ*-located words), and we apply this theory, exploiting the Budak-Işik-Pym...

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