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Magic numbers in metric spaces
Skálová, Alena ; Lukeš, Jaroslav (advisor) ; Veselý, Jiří (referee)
The main topic of this work is the Gross Theorem and its generalization - the Stadje Theorem. According to the Gross Theorem, for every compact connected metric space (X, d) there exists a unique magic number a(X, d) with the following property: For every finite set K ⊂ X there exists a point y ∈ X such that the average distance from y to K is a(X, d). The Stadje Theorem takes any real-valued continuous symmetric function f on X × X instead of a metric d. In this work we give a proof both of the existence and the uniqueness of the magic number from the Stadje Theorem. Examples of magic numbers in some particular metric spaces are presented. We also study the range of values which can a magic number attain, given some restrictions on the function f or on the space X. 1
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Orthogonality in Banach spaces
Mašková, Alice ; Milota, Jaroslav (referee) ; Lukeš, Jaroslav (advisor)
In the present work we study properties of orthogonality in Hilbert spaces and possibilities of extending definition to more general type of spaces, Banach spaces. We concentrate mostly on Birkhoff-James orthogonality and investigate, which properties of Hilbert space orthogonality are still valid for Banach spaces, otherwise we provide counter-examples. As the orthogonality is generally not symmetric, we have to distinguish between right and left properties. We use Birkhoff-James orthogonality to characterize smooth and strictly convex Banach spaces. Then we study properties of Hilbert space orthogonal projection and its generalizations for Banach spaces.We study projections of norm equal one and minimal projections.
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Isomorphic properties of spaces of continuous affine functions
Ludvík, Pavel ; Lukeš, Jaroslav (referee) ; Spurný, Jiří (advisor)
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 ; Malý, Jan (referee) ; Lukeš, Jaroslav (advisor)
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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