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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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Separable reduction theorems, systems of projections and retractions
Cúth, Marek ; Kalenda, Ondřej (advisor) ; Kubiš, Wieslaw (referee) ; Spurný, Jiří (referee)
This thesis consists of four research papers. In the first paper we study whether certain properties of sets (functions) are separably determined. In our results we use the "method of elementary submodels". In the second paper we generalize some results concerning Valdivia compacta (equivalently spaces with a commutative retractional skeleton) to the context of spaces with a retractional skeleton (not necessarily commutative). The third paper further studies the structure of spaces with a projectional (resp. retractional) skeleton. Under certain conditions we prove the existence of a "simultaneous projectional skeleton" and we use this result to prove other statements concerning the structure of spaces with a projectional (resp. retractional) skeleton. In the last paper we study the method of elementary submodels in a greater detail and we compare it with the "method of rich families". 1
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Exceptional Sets in Mathematical Analysis
Rmoutil, Martin ; Kalenda, Ondřej (advisor)
Title: Exceptional Sets in Mathematical Analysis Author: Martin Rmoutil Department: Department of Mathematical Analysis Supervisor: Doc. RNDr. Ondřej Kalenda, Ph.D., DSc., Department of Mathematical Analysis Abstract: The present thesis consists of four research articles. In the first paper we study the notion of σ-lower porous set; our main result is the existence of two closed sets A, B ⊂ R which are not σ-lower porous, but their product in R2 is lower porous. In the second and third article we use a set-theoretical method of el- ementary submodels involving the Lwenheim-Skolem theorem to prove that certain σ-ideals of sets in Banach spaces are separably determined. In the second article we do so for σ-porous sets and σ-lower porous sets. In the next article we refine these methods obtaining separable determination of a wide class of σ-ideals. In both cases we derive interesting corollaries which extend known theorems in separable spaces to the nonseparable setting; for example, we obtain the following theorem. Any continuous convex function on an Asplund space is Frchet differentiable outside a cone small set. In the fourth article we introduce the following notion. A closed set A ⊂ Rd is said to be c-removable if the following is true: Every real function on Rd is convex whenever it is continuous on Rd...
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Descriptive and topological aspects of Banach space theory
Kurka, Ondřej
of doctoral thesis Descriptive and topological aspects in Banach space theory Deskriptivní a topologické aspekty v teorii Banachových prostorů Ondřej Kurka 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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Quantitative properties of Banach spaces
Krulišová, Hana ; Kalenda, Ondřej (advisor)
The present thesis consists of four research papers. Each article deals with quan- tifications of certain properties of Banach spaces. The first paper is devoted to the Grothendieck property. The main result is that the space ∞ enjoys its quan- titative version. The second paper investigates quantifications of the Banach- Saks and the weak Banach-Saks property. The relationship of compact, weakly compact, Banach-Saks, and weak Banach-Saks sets is quantified, as well as some characterizatons of weak Banach-Saks sets. In the third article we discuss possible quantifications of Pelczy'nski's property (V), their characterizations and relations to quantitative versions of other properties of Banach spaces. The last paper is a continuation of the third one. We prove that C∗ -algebras have a quantita- tive version of the property (V), which generalizes one of the results obtained in the previous paper. Moreover, we establish a relationship between quantita- tive versions of the property (V) and the Grothendieck property in dual Banach spaces. 1
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Lipschitz-free spaces
Langr, Ondřej ; Cúth, Marek (advisor) ; Johanis, Michal (referee)
In this work we deal with basic properties of Lipschitz-free space. In the first part we especially show how these spaces are constructed and we show that they are characterized by "Universal property". In the second part we give an explicit formula for the calculation of the norm of an element in the general Lipschitz- free space over metric space containing four points. It looks that this formula is nowhere published, therefore this is probably the original result of this work. 1
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Quantitative properties of Banach spaces
Krulišová, Hana ; Kalenda, Ondřej (advisor) ; Raja Baño, Matias (referee) ; Hamhalter, Jan (referee)
The present thesis consists of four research papers. Each article deals with quan- tifications of certain properties of Banach spaces. The first paper is devoted to the Grothendieck property. The main result is that the space ∞ enjoys its quan- titative version. The second paper investigates quantifications of the Banach- Saks and the weak Banach-Saks property. The relationship of compact, weakly compact, Banach-Saks, and weak Banach-Saks sets is quantified, as well as some characterizatons of weak Banach-Saks sets. In the third article we discuss possible quantifications of Pelczy'nski's property (V), their characterizations and relations to quantitative versions of other properties of Banach spaces. The last paper is a continuation of the third one. We prove that C∗ -algebras have a quantita- tive version of the property (V), which generalizes one of the results obtained in the previous paper. Moreover, we establish a relationship between quantita- tive versions of the property (V) and the Grothendieck property in dual Banach spaces. 1
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Quantitative properties of Banach spaces
Krulišová, Hana ; Kalenda, Ondřej (advisor)
The present thesis consists of four research papers. Each article deals with quan- tifications of certain properties of Banach spaces. The first paper is devoted to the Grothendieck property. The main result is that the space ∞ enjoys its quan- titative version. The second paper investigates quantifications of the Banach- Saks and the weak Banach-Saks property. The relationship of compact, weakly compact, Banach-Saks, and weak Banach-Saks sets is quantified, as well as some characterizatons of weak Banach-Saks sets. In the third article we discuss possible quantifications of Pelczy'nski's property (V), their characterizations and relations to quantitative versions of other properties of Banach spaces. The last paper is a continuation of the third one. We prove that C∗ -algebras have a quantita- tive version of the property (V), which generalizes one of the results obtained in the previous paper. Moreover, we establish a relationship between quantita- tive versions of the property (V) and the Grothendieck property in dual Banach spaces. 1
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Exceptional Sets in Mathematical Analysis
Rmoutil, Martin ; Kalenda, Ondřej (advisor)
Title: Exceptional Sets in Mathematical Analysis Author: Martin Rmoutil Department: Department of Mathematical Analysis Supervisor: Doc. RNDr. Ondřej Kalenda, Ph.D., DSc., Department of Mathematical Analysis Abstract: The present thesis consists of four research articles. In the first paper we study the notion of σ-lower porous set; our main result is the existence of two closed sets A, B ⊂ R which are not σ-lower porous, but their product in R2 is lower porous. In the second and third article we use a set-theoretical method of el- ementary submodels involving the Lwenheim-Skolem theorem to prove that certain σ-ideals of sets in Banach spaces are separably determined. In the second article we do so for σ-porous sets and σ-lower porous sets. In the next article we refine these methods obtaining separable determination of a wide class of σ-ideals. In both cases we derive interesting corollaries which extend known theorems in separable spaces to the nonseparable setting; for example, we obtain the following theorem. Any continuous convex function on an Asplund space is Frchet differentiable outside a cone small set. In the fourth article we introduce the following notion. A closed set A ⊂ Rd is said to be c-removable if the following is true: Every real function on Rd is convex whenever it is continuous on Rd...
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Separable reduction theorems, systems of projections and retractions
Cúth, Marek ; Kalenda, Ondřej (advisor) ; Kubiš, Wieslaw (referee) ; Spurný, Jiří (referee)
This thesis consists of four research papers. In the first paper we study whether certain properties of sets (functions) are separably determined. In our results we use the "method of elementary submodels". In the second paper we generalize some results concerning Valdivia compacta (equivalently spaces with a commutative retractional skeleton) to the context of spaces with a retractional skeleton (not necessarily commutative). The third paper further studies the structure of spaces with a projectional (resp. retractional) skeleton. Under certain conditions we prove the existence of a "simultaneous projectional skeleton" and we use this result to prove other statements concerning the structure of spaces with a projectional (resp. retractional) skeleton. In the last paper we study the method of elementary submodels in a greater detail and we compare it with the "method of rich families". 1
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