Unit:
Κατεύθυνση Θεωρητικά ΜαθηματικάLibrary of the School of Science
Supervisors info:
1. Επιβλέπων: Παντελεήμων Δοδός - Ντοντός, Αναπληρωτής Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ,
2. Κωνσταντίνος Τύρος, Αναπληρωτής Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ,
3. Δημήτρης Χελιώτης, Αναπληρωτής Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ
Original Title:
APPLICATIONS OF COMBINATORIES IN FUNCTIONAL THEORY: ROSENTHAL'S THEOREM
Translated title:
APPLICATIONS OF COMBINATORIES IN FUNCTIONAL THEORY: ROSENTHAL'S THEOREM
Summary:
The current thesis is targeted on revealing the connection between the results and concepts of Combinatories and Descriptive Set Theory and the field of Functional Analysis. Descriptive set theory is the study of the so-called “definable” subsets of Polish topological spaces, i.e. subsets that are defined explicitly from the very basic sets (the open sets) using simple set-theoretic operations such as complementation, countable unions and projections, and is one of the main areas of research in set theory.
The main goal of the current thesis is to present the proof of Rosenthal's theorem, one of the most remarkable results in Banach spaces' geometry, which entails a both necessary and sufficient condition for a bounded sequence in a Banach space X to have a weakly Cauchy subsequence. The essential condition is related to the geometry of the space and requires that no isomorphic copy of l^{1} can be embedded in X.
More precisely, using as trigger the result of Eberlein-Smulian, it is indicated that in order to be able to extract from each bounded sequence in X a weakly convergent subsequence a both necessary and sufficient condition for X is to be reflexive. Reducing the request to the existence of a weakly Cauchy subsequence, the challenge is focused on the examination of non-reflexive Banach spaces, since the above result already provides a direction for the reflexive ones.
In fact, considering a separable Banach space X and assuming X^{*} also being separable, we notice that the bounded sequences in X have weakly Cauchy subsequences. This is easy to be proved considering a countable dense sequence (d_{n}) in the unit ball X^{*} and the images of the bounded (x_{n}) sequence in X through them. Precisely, for d_{1} the sequence (d_{1}x_{n}) is bounded in reals \mathbb{R}, thus it has a convergent subsequence, let (d_{1}x_{n}^{1}). Now we apply the operator d_{2} in the (x_{n}^{1}) and using the same argument we successively are led to the determination of a subsequence of the bounded (x_{n}), with the requested property. However this might not always be feasible. For example if we consider (e_{n}) being the n-th unit vector in l^{1} then (e_{n}) has no weakly Cauchy subsequence.
Rosenthal proved that the above counterexample is, in essense, the only one. Nevertheless the most interesting fact is that the proof of Rosenthal's theorem itself was established in combinatory results, extending one of the most intutive facts of combinatory theory, known as the Pigeon Hole Principle.
The current work consists of three parts in which are introduced the essential concepts and results of the combined fields, as per above, enabling us to prove the Main Result in the last chapter.
More precisely, in Part I is introduced one of the basic structural tools of Descriptive set theory, the “trees”, which is used in constructive proofs in the later sections. In Part II we summarise the basic concepts and structures related to Polish spaces and Baire spaces, highlighting the homeomorphic behaviour of their subspaces with characteristic spaces such as Cantor \mathcal{C} and Baire space \mathcal{N}. Finally, Part III is focused on describing the combinatory Partition Theorems along with their extensions in the examination of Polish spaces' partitions in “definable” subsets and the partitions of [\mathbb{N}]^{\aleph}. Rosenthal's theorem is an application of the said Partition Theorems and is presented as a conclusion in the current thesis.
Main subject category:
Science
Keywords:
Rosenthal theorem,Combinatories, Descriptive set theory,Functional analysis, Cantor, Ellentuck topology,Ramsey theorem, pigeon hole, Polish spaces
File:
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Master Thesis_Rosenthal Theorem_Maria Tzanadami.pdf
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