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Sigma-porous sets and the differentiation theory
Koc, Martin ; Zajíček, Luděk (advisor) ; Zelený, Miroslav (referee) ; Kolář, Jan (referee)
of the dissertation thesis Title: Sigma-porous sets and the differentiation theory Author: Martin Koc Department: Department of mathematical analysis Supervisor: Prof. RNDr. Luděk Zajíček, DrSc., Department of mathematical analysis Abstract: The thesis consists of five research articles. In the first one, it is shown that there exists a closed upper porous (in a strong sense) subset of a nonempty, topolo- gically complete metric space without isolated points that is not σ-lower porous (in a weak sense). In the second article, a new notion of porosity with respect to a measure, that generalizes the upper porosity of a measure, is introduced. Several natural definitions of this notion are investigated. The main result of this chapter is a decomposition theorem for sets that are σ-porous with respect to a measure. The third article deals with sets of points at which arbitrary real functions are Lipschitz from one side and not Lipschitz from another side. A full characterization of the system generated by sets of this type is proved. In the fourth article, several results on relations among metric derived numbers for functions with values in metric spaces are shown. The last chapter deals with existence of differentiable extensions for functions defined on closed subsets of Rn . Its main result...
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Perfect functions of the first Baire class
Skovajsa, Břetislav ; Spurný, Jiří (advisor) ; Zajíček, Luděk (referee)
A wide class of problems in mathematical analysis can be described as searching for properties P such that for each F from a given system of mappings F between spaces K and L an arbitrary real valued function on L has the property P if and only if its composition with F also has this property. The inspiration for this text comes from [1], where the mentioned problem is examined in the form of stability of Baire classes of functions towards composition with a continuous mapping between compact topological spaces. The goal of this text will be to get acquainted with the original result, to slightly improve it on compact metric spaces, then to take a closer look at the finer structure of B1 functions and to try to find a similar kind of stability in this environment. [1] J. Lukeš, J. Malý, I. Netuka, J. Spurný, Integral representation theory: ap- plications to convexity, Banach spaces and potential theory, Walter de Gruyter (2010).
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Descriptive properties of sets in Banach spaces
Kurka, Ondřej ; Holický, Petr (advisor) ; Zajíček, Luděk (referee)
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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Properties of sigma-porous sets
Rmoutil, Martin ; Zajíček, Luděk (advisor) ; Zelený, Miroslav (referee)
In the present thesis we prove several new results concerning -porous sets. In the first two chapters we examine some properties of related sets in the space R while in the third chapter we concentrate on an entirely different problem formulated in the setting of topologically complete metric spaces. To be more specific, in the first chapter we prove non- -porosity of the set Ad of all real numbers x (0, 1) with decimal expansion containing the number 9 with density d. In spite of being relatively difficult, this new result has little importance in itself. It merely answers a natural question which arises from an article of L. Zajíček [8]. The main result presented in the second chapter is a significant improve- ment of the following result of R.J. Najáres and L. Zajíček from the article [5]: There exists a closed set F R which is right porous, but is not -left porous. Thus for any kind of "upper" porosity (i.e. a porosity defined using limsup) it is now even more unlikely for any connection between "left" and "right" to be discovered. From another work [10] of L. Zajíček arises the following question: If A X and B Y are two non- -lower porous G -subsets of topologically complete metric spaces X and Y , is it necessarily true that the Cartesian product A × B is also non- -lower porous? The article [10]...
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