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Convergence in Banach Spaces
Silber, Zdeněk ; Kalenda, Ondřej (advisor) ; Plebanek, Grzegorz (referee) ; Cúth, Marek (referee)
The thesis consists of three articles. The common theme of the first two articles is the possibility of iterating weak∗ derived sets in dual Banach spaces. In the first article we prove that in the dual of any non-reflexive Banach space we can always find a convex set of order n for any n ∈ N, and a convex set of order ω +1. This result extends Ostrovskii's characterization of reflexive spaces as those spaces for which weak∗ derived sets coincide with weak∗ closures for convex sets. In the second article we prove an iterated version of another result of Ostrovskii, that a dual to a Banach space X contains a subspace whose weak∗ derived set is proper and norm dense, if and only if X is non-quasi-reflexive and contains an infinite-dimensional subspace with separable dual. In the third article we study quantitative results concerning ξ-Banach-Saks sets and weak ξ-Banach-Saks sets. We provide quantitative analogues to characterizations of weak ξ-Banach-Saks sets using ℓξ+1 1 spreading models and a quantitative version of the relation of ξ-Banach-Saks sets, weak ξ-Banach-Saks sets, norm compactness and weak compactness. We use these results to define a new measure of weak non-compactness and finally give some relevant examples. 1
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Ulam's problem
Kučerová, Tereza ; Cúth, Marek (advisor) ; Kalenda, Ondřej (referee)
In this bachelor's thesis we deal with Ulam's problem. In the first chapter, we introduce the basic definitions and the axiomatic theory of ZF extended by the Axiom of Choice; we also formulate and prove the Lemma that will be used for the proofs in the second and third chapters. In the second and third chapters, we prove that, assuming the Continuum Hypothesis holds, the Ulam's problem has a positive solution, and assuming the Full Measure Extension Axiom holds, the Ulam's problem has a negative solution. We carry out both proofs with a high degree of detail. Finally, in chapter four, we prove that the generalized Ulam's problem for sets with cardinality greater than that of the real numbers always has a negative solution. 1
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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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Composition operators on function spaces
Novotný, Matěj ; Spurný, Jiří (advisor) ; Kalenda, Ondřej (referee)
Univerzita Karlova Abstract of the bachelor thesis Composition operators on function spaces Matěj Novotný, Praha 2011 In the thesis we define what is an composition operator on the space of continuous or measurable functions of one complex variable so that we may proceed to study its properties depending on properties of the mapping the operator is induced by. We search for conditions under which the operator is continuous, compact and an isomorphism. We roughly estimate the spectrum of an operator defined on a space of continuous functions. 1
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Separable reduction theorems in functional analysis
Cúth, Marek ; Kalenda, Ondřej (advisor) ; Holický, Petr (referee)
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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Spaces of continuous functions with the pointwise topology
Slavata, Martin ; Spurný, Jiří (advisor) ; Kalenda, Ondřej (referee)
Title: Spaces of continuous functions with the pointwise topology Author: Martin Slavata Department: Department of Mathematical Analysis Supervisor: doc. RNDr. Jiří Spurný, Ph.D. Supervisor's e-mail address: Jiri.Spurny@mff.cuni.cz Abstract: This thesis describes properties of spaces of continuous functions with the topology of pointwise convergence. Emphasis is put on characterizations of compact subsets of such spaces and on compactness of the spaces themselves. The thesis describes properties of the class of angelic spaces (notion by Fremlin) and shows when spaces of continuous functions with pointwise topology belong to this class (result by J. Orihuela). Thus a generalization of a theorem of Grothendieck is obtained. Also a limitation of the class of angelic spaces is shown - it is not closed under topological product. This leads to the next topic of the thesis, the class of strictly angelic spaces (introduced by W. Govaerts) and its intersection with the class of spaces of continuous functions with pointwise topology. In the end the thesis shows under which conditions the space of continuous functions satisfies the definition of the respective notions related to compactness. Keywords: spaces of continuous functions; pointwise convergence; compactness; angelicity
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Some results in convexity and in Banach space theory
Kraus, Michal ; Lukeš, Jaroslav (advisor) ; Kalenda, Ondřej (referee) ; Smith, Richard (referee)
This thesis consists of four research papers. In the first paper we construct nonmetrizable compact convex sets with pathological sets of simpliciality, show- ing that the properties of the set of simpliciality known in the metrizable case do not hold without the assumption of metrizability. In the second paper we construct an example concerning remotal sets, answering thus a question of Martín and Rao, and present a new proof of the fact that in every infinite- dimensional Banach space there exists a closed convex bounded set which is not remotal. The third paper is a study of the relations between polynomials on Banach spaces and linear identities. We investigate under which conditions a linear identity is satisfied only by polynomials, and describe the space of poly- nomials satisfying such linear identity. In the last paper we study the coarse and uniform embeddability between Orlicz sequence spaces. We show that the embeddability between two Orlicz sequence spaces is in most cases determined only by the values of their upper Matuszewska-Orlicz indices. 1
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