
Images of Typical Continuous Functions
Nešvera, Michal ; Vejnar, Benjamin (advisor) ; Holický, Petr (referee)
It follows from the Baire theorem that comeagre sets in complete metric spaces are "topologically large". Properties that are satisfied by a large set are called typical. The proofs of statements concerning typical properties of continuous functions are the main part of this work. For this purpose, the necessary definitions are introduced in the first chapter and the completeness of spaces of continuous functions is proved. As the first example of a typical property, in the second chapter we prove the BanachMazurkiewicz theorem, which states that nondifferentiability is a typical property. The third chapter of this thesis is devoted to the study of typical properties of continuous mappings of the unit interval into the plane. In the last chapter, statements regarding the typical properties of continuous mappings of the unit interval into Euclidean spaces of higher dimensions are proved. 1


Proofs of Tychonoff's Theorem
Dvořáková, Johana ; Cúth, Marek (advisor) ; Holický, Petr (referee)
This bachelor thesis is devoted to four different proofs of Tychonoff's Theorem. The first proof is based on definitions of compact topological space and product topology. The second proof is a construction of convergent subnet of an arbitrary net in a product of compact spaces. The third proof uses the fact that topological space is compact if and only if every universal net is convergent. The last proof is based on characterization of compact spaces using systems of closed subsets with the finite intersection property. 1


New measures of weak noncompactness
Bendová, Hana ; Kalenda, Ondřej (advisor) ; Holický, Petr (referee)
The main topic of this thesis is the measures of weak noncompactness, which, in different ways, measure weak noncompactness of bounded sets in Banach spa ces. Besides some known measures of weak noncompactness, we introduce new measures, that are more natural in some sense, and we show the relationships be tween them. We prove quantitative versions of EberleinGrothendieck, Eberlein Šmulian, and James' theorems. Afterwards, we deal with measures of weak noncompactness of the unit ball and measures of weak noncompactness 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.


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.


Applications of descriptive set theory in mathematical analysis
Doležal, Martin ; Zelený, Miroslav (advisor) ; Holický, Petr (referee) ; Zapletal, Jindřich (referee)
We characterize various types of σporosity via an infinite game in terms of winning strategies. We use a modification of the game to prove and reprove some new and older in scribing theorems for σideals of σporous type in locally compact metric spaces. We show that there exists a closed set which is σ(1 − ε)symmetrically porous for every 0 < ε < 1 but which is not σ1symmetrically porous. Next, we prove that the realizable by an action unitary representations of a finite abelian group Γ on an infinitedimensional complex Hilbert space H form a comeager set in Rep(Γ, H). 1

 

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 nonBorel set of Fréchet subdi erentiability on every nonreflexive 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 nonre exive Banach space can be renormed not to have Borel set of normattaining functionals is shown. A characterization of nonquasire exive Banach spaces is given.


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 nonreflexive Banach space, equivalent strictly convex norms with the set of normattaining 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 Pclass 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


RadonNikodým compact spaces
Cepák, Jiří ; Spurný, Jiří (advisor) ; Holický, Petr (referee)
In the present work we study RadonNikodým 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.

 