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1
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Characterizations of elements of a double dual Banach space and their canonical reproductions

100%
Farmaki V.
Studia Mathematica
|
1993
|
tom 104
|
nr 1
61-74
EN
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 $(x^{(2n)})$ equivalent to the usual basis of $ℓ^1$ (resp. as the elements with the sequence $(x^{(4n-2)} - x^{(4n)})$ equivalent to the usual basis of $c_0$). Also, by analogous conditions but of isometric nature, we characterize the embeddability of $ℓ^1$ (resp. $c_0$) in X.
2
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On spreading $c_0$-sequences in Banach spaces

100%
Farmaki V.
Studia Mathematica
|
1999
|
tom 135
|
nr 1
83-102
EN
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) A Banach space X has the spreading-(s) property if and only if for every subspace Y of X and for every pair of sequences (x_n) in Y and $(x*_n)$ in Y*, with(x_n) weakly null in Y and $(x_n*)$ uniformly weakly null in Y* (in the sense of Mercourakis), we have $x*_n(x_n) → 0$ (i.e. X has a hereditary weak Dunford-Pettis property). (b) A Banach space X has the spreading-(u) property if and only if $B_1(X) ⊆ B_{1/4}(X)$ in the sense of the classification of Baire-1 elements of X** according to Haydon-Odell-Rosenthal. (c) The spreading-(s) property implies the spreading-(u) property. Result (c), proved via infinite combinations, connects an internal condition on a Banach space with an external one.
3
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Ordinal indices and Ramsey dichotomies measuring c₀-content and semibounded completeness

100%
Farmaki V.
Fundamenta Mathematicae
|
2002
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tom 172
|
nr 2
153-179
EN
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 $ξ^{(χₙ)}₀$ and the semibounded completeness index $ξ^{(χₙ)}_b$, and we examine their relationship. The countable ordinal values that these indices can take are always of the form $ω^{ζ}$. These results extend, to the countable ordinal level, an earlier result by Odell, which was stated only for the limiting case of the first uncountable ordinal.
4
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Extended Ramsey theory for words representing rationals

63%
Farmaki V., Koutsogiannis A.
Fundamenta Mathematicae
|
2013
|
tom 223
|
nr 1
1-27
EN
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 representation of rational numbers, to obtain an analogous partition theory for the set of rational numbers.
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