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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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Riemann type integral in Banach spaces
Mrhal, Filip ; Lukeš, Jaroslav (advisor) ; Zajíček, Luděk (referee)
Title: Riemann type integral in Banach spaces Author: Filip Mrhal Department: Department of Mathematical Analysis Supervisor: Prof. RNDr. Jaroslav Lukeš, DrSc., Department of Mathematical Analysis Abstract: In this thesis we study some differences in the behaviours of the Ri- emann integral when integrating functions from any compact subinterval of real numbers to real numbers or to any Banach space. Especially, we outline that the Lebesgue theorem is no longer valid in relationship to functions with images in some Banach spaces. We show that for some well-known Banach spaces using counterexamples. Keywords: Riemann integral, Banach space, Lebesgue theorem 1
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Isomorphic properties of spaces of continuous affine functions
Ludvík, Pavel ; Spurný, Jiří (advisor) ; Lukeš, Jaroslav (referee)
The thesis deals with Banach-Stone theorem, its modi cations and generalizations. The preface of the thesis contains a lot of well known results and useful assertions from such elds of mathematics as measury theory, functional analysis, topology and most importantly convex analysis. The second chapter pursues proofs of classical Banach-Stone theorem and Eilenberg theorem, which works in another context than the original theorem. Chapter number three contains contribution of A. Lazar, who proved variation of Banach-Stone theorem for afine functions on simplexes. The chapter follows with generalizations of his results and it is closed with our own slight generalization. The last chapter pays attention to "almost isometries". The chapter comes out from theorem proved by A. Amir and continues with improvements achieved by H.B. Cohen and C.-H. Chu. The last part includes our own contribution to the subject.
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Integral representation theorems in noncompact cases
Kraus, Michal ; Lukeš, Jaroslav (advisor) ; Malý, Jan (referee)
Classical Choquet's theory deals with compact convex subsets of locally convex spaces. This thesis discuss some aspects of generalization of Choquet's theory for a broader class of sets, for example those which are assumed to be only closed and bounded instead of compact. Because Radon measures are usually defined for locally compact topological spaces, and this is not the case of the closed unit ball in a Banach space of infinite dimension, there are used the so called Baire measures in this setting. This thesis particularly deals with the question of existence of resultants of these measures, with the properties of the resultant map, with the analogy of Bauer's characterization of extreme points and with some other concepts known from compact theory. By using some examples we show that many of these theorems doesn't hold in noncompact setting. We also mention forms of these theorems which can be proved.
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Baire and Harmonic Functions
Pošta, Petr ; Lukeš, Jaroslav (advisor)
Title: Baire and Harmonic Functions Author: Petr Pošta Department: Department of Mathematical Analysis Supervisor: prof. RNDr. Jaroslav Lukeš, DrSc., Department of Mathematical Analysis Abstract: The present thesis consists of six research papers. The first four articles deal with topics related to potential theory, Baire-one functions and its important subclasses, in particular differences of semicontinuous functions. The first paper is devoted to the stability of the Dirichlet problem for which a new criterion in terms of Poisson equation is provided. The second paper improves the recent result obtained by Lukeš et al. It shows that the classical Dirichlet solution belongs to the B1/2 subclass of Baire-one functions. A generalization of this result to the abstract context of the Choquet theory on functions spaces is provided. Finally, an abstract Dirichlet problem for the boundary condition belonging to the class of differences of semincontinuous functions is discussed. The third paper concentrates on the Lusin-Menshov property and the approximation of Baire- one and finely continuous functions by differences of semicontinuous and finely continuous functions. It provides an exposition of topologies (various density topologies as well as the fine topologies in both linear and non-linear potential...
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