Ramsey Methods in Analysis

Argyros, Spiros A.; Todorčević, Stevo

doi:10.1007/3-7643-7360-1

Cited by 92 publications

(120 citation statements)

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“…sequences and the basic inequality. Actually it is identical with the arguments presented in Section 2.2 of [7].…”

Section: Lemma 50mentioning

confidence: 61%

“…For the proof we refer to [7,Lemma 2.22] and [6,Lemma 4.6]. The following result has its roots in Schlumprecht's paper [51].…”

Section: Lemma 49mentioning

confidence: 90%

“…Here we want to comment the use of the special j -block sequences in the definition of dependent sequences. A key ingredient for showing inequality (17) is a tree-like property satisfied by the (n 2l−1 )-special sequences (see [7,Proposition 3.3]). When we deal with norms on c 00 (N) then the tree-like property is also satisfied by all restrictions of the special sequences on intervals of N. However this is not valid when we deal with c 00 (N × N) and we consider restrictions on rectangles generated by intervals of N. Notice that this problem disappears if we consider special j -block sequences and this is the reason why we introduced this concept.…”

Section: Remark 10mentioning

confidence: 99%

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Genericity and amalgamation of classes of Banach spaces

Argyros

Dodos

2007

Advances in Mathematics

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We study universality problems in Banach space theory. We show that if A is an analytic class, in the Effros-Borel structure of subspaces of C([0, 1]), of non-universal separable Banach spaces, then there exists a non-universal separable Banach space Y , with a Schauder basis, that contains isomorphs of each member of A with the bounded approximation property. The proof is based on the amalgamation technique of a class C of separable Banach spaces, introduced in the paper. We show, among others, that there exists a separable Banach space R not containing L 1 (0, 1) such that the indices β and r ND are unbounded on the set of Baire-1 elements of the ball of the double dual R * * of R. This answers two questions of H.P. Rosenthal.We also introduce the concept of a strongly bounded class of separable Banach spaces. A class C of separable Banach spaces is strongly bounded if for every analytic subset A of C there exists Y ∈ C that contains all members of A up to isomorphism. We show that several natural classes of separable Banach spaces are strongly bounded, among them the class of non-universal spaces with a Schauder basis, the class of reflexive spaces with a Schauder basis, the class of spaces with a shrinking Schauder basis and the class of spaces with Schauder basis not containing a minimal Banach space X.

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“…sequences and the basic inequality. Actually it is identical with the arguments presented in Section 2.2 of [7].…”

Section: Lemma 50mentioning

confidence: 61%

“…For the proof we refer to [7,Lemma 2.22] and [6,Lemma 4.6]. The following result has its roots in Schlumprecht's paper [51].…”

Section: Lemma 49mentioning

confidence: 90%

Section: Remark 10mentioning

confidence: 99%

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Genericity and amalgamation of classes of Banach spaces

Argyros

Dodos

2007

Advances in Mathematics

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“…One could not expect a similar result for an arbitrary uncountable family as above. Indeed, there exists an uncountable weakly discrete family accumulating to 0 in a reflexive and HI saturated space X (see [6]). …”

Section: Theorem 3 Let X Be a Separable Banach Space Andmentioning

confidence: 99%

“…For a presentation of some of the partition theorems we use, we refer the reader to [6]. Applications of Ramsey theory for trees in Analysis and Topology can be found in [3,38].…”

Section: The Ramsey Theoretical Backgroundmentioning

confidence: 99%