Weakly null sequences with an unconditional subsequence

Arvanitakis, Alexander D.

doi:10.1090/s0002-9939-05-07948-7

Cited by 4 publications

(9 citation statements)

References 7 publications

(6 reference statements)

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“…The following result, also proved in , concerns the case where the space X in the above theorem is finite dimensional. Theorem Let K be a Hausdorff compact space and

{(f_{n})}_{n \in double-struckN}

f_{n} : K \to {double-struckR}^{m}

, a uniformly bounded sequence of continuous functions which converges pointwise to zero.…”

Section: Introductionmentioning

confidence: 84%

“…Finally, the combinatorial proof of Rosenthal's theorem has been expanded and some stronger results have been obtained. As pointed out in , Theorem can not be extended in the case where the range of

f_{n}

is a finite set of arbitrarily large cardinality. However, Arvanitakis expanded this theorem in the case where the cardinality of the range of

f_{n}

is finite and uniformly bounded by some positive integer.…”

Section: Introductionmentioning

confidence: 99%

“…As pointed out in , Theorem can not be extended in the case where the range of

f_{n}

is a finite set of arbitrarily large cardinality. However, Arvanitakis expanded this theorem in the case where the cardinality of the range of

f_{n}

is finite and uniformly bounded by some positive integer. Theorem Let K be a Hausdorff compact space, X a Banach space and

{(f_{n})}_{n \in double-struckN}

f_{n} : K \to X

, a normalized sequence of continuous functions.…”

Section: Introductionmentioning

confidence: 99%

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Some combinatorial principles for trees and applications to tree families in Banach spaces

Poulios

Tsarpalias

2014

Mathematical Logic Qtrly

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Suppose that (xs)s∈S is a normalized family in a Banach space indexed by the dyadic tree S. Using Stern's combinatorial theorem we extend important results from sequences in Banach spaces to tree‐families. More precisely, assuming that for any infinite chain β of S the sequence (xs)s∈β is weakly null, we prove that there exists a subtree T of S such that for any infinite chain β of T the sequence (xs)s∈β is nearly (resp., convexly) unconditional. In the case where (fs)s∈S is a family of continuous functions, under some additional assumptions, we prove the existence of a subtree T of S such that for any infinite chain β of T, the sequence (fs)s∈β is unconditional. Finally, in the more general setting where for any chain β, (xs)s∈β is a Schauder basic sequence, we obtain a dichotomy result concerning the semi‐boundedly completeness of the sequences (xs)s∈β.

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“…The following result, also proved in , concerns the case where the space X in the above theorem is finite dimensional. Theorem Let K be a Hausdorff compact space and

{(f_{n})}_{n \in double-struckN}

f_{n} : K \to {double-struckR}^{m}

, a uniformly bounded sequence of continuous functions which converges pointwise to zero.…”

Section: Introductionmentioning

confidence: 84%

f_{n}

is a finite set of arbitrarily large cardinality. However, Arvanitakis expanded this theorem in the case where the cardinality of the range of

f_{n}

is finite and uniformly bounded by some positive integer.…”

Section: Introductionmentioning

confidence: 99%

“…As pointed out in , Theorem can not be extended in the case where the range of

f_{n}

is a finite set of arbitrarily large cardinality. However, Arvanitakis expanded this theorem in the case where the cardinality of the range of

f_{n}

is finite and uniformly bounded by some positive integer. Theorem Let K be a Hausdorff compact space, X a Banach space and

{(f_{n})}_{n \in double-struckN}

f_{n} : K \to X

, a normalized sequence of continuous functions.…”

Section: Introductionmentioning

confidence: 99%

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Some combinatorial principles for trees and applications to tree families in Banach spaces

Poulios

Tsarpalias

2014

Mathematical Logic Qtrly

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“…An infinite-dimensional Banach space contains either an infinite unconditional basic sequence or a hereditarily indecomposable Banach space. 3 Concerning this result we note that the space of Theorem 1.1 is actually hereditarily indecomposable while the space of Theorem 1.2 being reflexive and non-separable must have many decompositions as sum of two closed infinite-dimensional subspaces. In this article, we shall discuss the following two general versions of the problem.…”

Section: Theorem 13 ([16]mentioning

confidence: 97%

“…The classical procedure of Mazur for selecting a Schauder basic sequence inside an arbitrary Banach space when applied to subsequences of a given weakly null sequence gives us the following result of Bessaga and Pelczynski [5]. 3 Recall that an infinite-dimensional Banach space X is indecomposable if it cannot be written as sum Y Z of two closed infinite-dimensional subspaces Y and Z. We say that X is hereditarily indecomposable if all closed infinite-dimensional subspaces of X are indecomposable.…”

Section: Finite and Partial Unconditionalitymentioning

confidence: 99%

Ramsey-theoretic analysis of the conditional structure of weakly-null sequences

Todorčević¹

European Congress of Mathematics Kraków, 2 – 7 July, 2012

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Understanding the possible conditional structure in a given weakly-null sequence (xi) in some normed space X lies in the heart of several classical problems of this area of mathematics. We will expose the set-theoretic and Ramsey-theoretic methods relevant to both the lack and the existence of this conditional structure. We will concentrate on more recent results and will point out problems for further study.

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Απειροσυνδυαστικές Μέθοδοι Σε Θέματα Συναρτησιακής Ανάλυσης Και Η Ιδιότητα Σταθερού Σημείου

Poulios¹,

Πούλιος²

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Η παρούσα διδακτορική διατριβή αποτελείται από δύο μέρη. Στο πρώτο μέρος, το οποίο αντιστοιχεί στα Κεφάλαια 1, 2 και 3 της εργασίας, καλύπτονται θέματα σχετικά με εφαρμογές της Θεωρίας Ramsey στη Συναρτησιακή Ανάλυση. Στο δεύτερο μέρος, που περιλαμβάνει τα Κεφάλαια 4 και 5, κατασκευάζεται μια σημαντική κλάση χώρων Banach και μελετάται η ιδιότητα σταθερού σημείου. Στο υπόλοιπο της εισαγωγής, ακολουθεί μια σύντομη περιγραφή των αποτελεσμάτων που περιέχονται στη διατριβή. […]

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Weakly null sequences with an unconditional subsequence

Cited by 4 publications

References 7 publications

Some combinatorial principles for trees and applications to tree families in Banach spaces

Some combinatorial principles for trees and applications to tree families in Banach spaces

Ramsey-theoretic analysis of the conditional structure of weakly-null sequences

Απειροσυνδυαστικές Μέθοδοι Σε Θέματα Συναρτησιακής Ανάλυσης Και Η Ιδιότητα Σταθερού Σημείου

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