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Collections of compact sets in descriptive set theory
Vlasák, Václav ; Zelený, Miroslav (advisor) ; Holický, Petr (referee) ; Tišer, Jaroslav (referee)
1 Title: Collections of compact sets in descriptive set theory Author: Václav Vlasák Department: Department of Mathematical Analysis Supervisor: Doc. RNDr. Miroslav Zelený, Ph.D. Author's e-mail address: vlasakmm@volny.cz Abstract: This work consists of three articles. In Chapter 2, we dissert on the connections between complexity of a function f from a Polish space X to a Polish space Y and complexity of the set C(f) = {K ∈ K(X); f K is continuous}, where K(X) denotes the space of all compact subsets of X equipped with the Vietoris topology. We prove that if C(f) is analytic, then f is Borel; and assuming ∆1 2-Determinacy we show that f is Borel if and only if C(f) is coanalytic. Similar results for projective classes are also presented. In Chapter 3, we continue in our investigation of collection C(f) and also study its restriction on convergent sequences (C(f)). We prove that C(f) is Borel if and only if f is Borel. Similar results for projective classes are also presented. The Chapter 4 disserts on HN -sets, which form an important subclass of the class of sets of uniqueness for trigonometric series. We investigate the size of these classes which is reflected by the family of measures called polar which annihilate all the sets belonging to the given class. The main aim of this chapter is to answer in...
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New measures of weak non-compactness
Bendová, Hana ; Kalenda, Ondřej (advisor) ; Holický, Petr (referee)
The main topic of this thesis is the measures of weak non-compactness, which, in different ways, measure weak non-compactness of bounded sets in Banach spa- ces. Besides some known measures of weak non-compactness, we introduce new measures, that are more natural in some sense, and we show the relationships be- tween them. We prove quantitative versions of Eberlein-Grothendieck, Eberlein- Šmulian, and James' theorems. Afterwards, we deal with measures of weak non-compactness of the unit ball and measures of weak non-compactness of sets in Banach spaces with w∗ -angelic dual unit ball. We prove that in these cases some of the defined measures coincide. Finally, we focus on the behaviour of the defined measures while passing to convex and absolute convex hull. We prove quantitative version of Krein's theorem and we also prove that most of the mea- sures do not change when passing to convex and absolute convex hull in Banach spaces with w∗ -angelic dual unit ball.
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Descriptive and topological aspects of Banach space theory
Kurka, Ondřej ; Holický, Petr (advisor) ; Fabian, Marián (referee) ; Hájek, Petr (referee)
The thesis consists of three papers of the author. In the first paper, it is shown that the sets of Fréchet subdifferentiability of Lipschitz functions on a Banach space X are Borel if and only if X is reflexive. This answers a ques- tion of L. Zajíček. In the second paper, a problem of G. Debs, G. Godefroy and J. Saint Raymond is solved. On every separable non-reflexive Banach space, equivalent strictly convex norms with the set of norm-attaining func- tionals of arbitrarily high Borel class are constructed. In the last paper, binormality, a separation property of the norm and weak topologies of a Ba- nach space, is studied. A result of P. Holický is generalized. It is shown that every Banach space which belongs to a P-class is binormal. It is also shown that the asplundness of a Banach space is equivalent to a related separation property of its dual space. 1
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Radon-Nikodým compact spaces
Cepák, Jiří ; Holický, Petr (referee) ; Spurný, Jiří (advisor)
In the present work we study Radon-Nikodým compact spaces (RN compacta for short) their topological characterizations and properties with emphasis on those related to the problem of continuous image of RN compact. First chapter consists of auxiliary results. In second chapter we give eight characterizations of RN compacta as well as several examples. In third chapter we introduce three notions weaker than that of RN compact and stable under continuous images and we show that they are equivalent. Last chapter is devoted to partial positive solutions to the problem of continuous image.
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Separable reduction theorems in functional analysis
Cúth, Marek ; Holický, Petr (referee) ; Kalenda, Ondřej (advisor)
In the presented work we are studying, whether some properties of sets (functions) can be separably reduced. It means, whether it is true, that a set (function) has given property if and only if it has this property in a special separable subspace, dependent only on the given set (function). We are interested in properties of sets "be dense, nowhere dense, meager, residual and porous" and in properties of functions "be continuous, semicontinuous and Fréchet di erentiable". Out method of creating separable subspaces enables us to combine our results, and so we easily get separable reductions of function properties such as "be continuous on a dense subset", "be Fréchet di erentiable on a residual subset", etc. Finally, we show some applications of presented separable reduction theorems, which enable us to show, that some propositions proven by Zajíček, Lindenstrauss and Preiss hold under other assumptions as well.
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Descriptive properties of sets in Banach spaces
Kurka, Ondřej ; Zajíček, Luděk (referee) ; Holický, Petr (advisor)
An essential part of the work is devoted to the study of the sets of Fréchet subdi erentiability from the view of the descriptive set theory. Proofs of the known results of L. Zajíek, P. Holický, M. Laczkovich and M. Šmídek are given. A new result is that there exists a Lipschitz function with non-Borel set of Fréchet subdi erentiability on every non-reflexive Banach space. The Borel classes of the sets of Fréchet subdi erentiability of continuous functions on reflexive spaces are studied as well. Further, some sets of sequences in Banach spaces are investigated. A modi ed proof of the theorem of R. Kaufman which says that every non-re exive Banach space can be renormed not to have Borel set of norm-attaining functionals is shown. A characterization of non-quasire exive Banach spaces is given.
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