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Topological and descriptive methods in the theory of function and Banach spaces
Kačena, Miroslav ; Spurný, Jiří (advisor) ; Netuka, Ivan (referee) ; Kalenda, Ondřej (referee)
The thesis consists of four research papers. The first three deal with the Choquet theory of function spaces. In Chapter 1, a theory on products and projective limits of function spaces is developed. It is shown that the product of simplicial spaces is a simplicial space. The stability of the space of maximal measures under continuous affine mappings is studied in Chapter 2. The third chapter employs results from the previous chapters to construct an example of a function space where the abstract Dirichlet problem is not solvable for any class of Baire-n functions with $n\in N$. It is shown that such an example cannot be constructed via the space of harmonic functions. In the final chapter, the recently introduced class of sequentially Right Banach spaces is being investigated. Connections to other isomorphic properties of Banach spaces are established and several characterizations are given.
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Spaces of continuous functions with the pointwise topology
Slavata, Martin ; Spurný, Jiří (advisor) ; Kalenda, Ondřej (referee)
Title: Spaces of continuous functions with the pointwise topology Author: Martin Slavata Department: Department of Mathematical Analysis Supervisor: doc. RNDr. Jiří Spurný, Ph.D. Supervisor's e-mail address: Jiri.Spurny@mff.cuni.cz Abstract: This thesis describes properties of spaces of continuous functions with the topology of pointwise convergence. Emphasis is put on characterizations of compact subsets of such spaces and on compactness of the spaces themselves. The thesis describes properties of the class of angelic spaces (notion by Fremlin) and shows when spaces of continuous functions with pointwise topology belong to this class (result by J. Orihuela). Thus a generalization of a theorem of Grothendieck is obtained. Also a limitation of the class of angelic spaces is shown - it is not closed under topological product. This leads to the next topic of the thesis, the class of strictly angelic spaces (introduced by W. Govaerts) and its intersection with the class of spaces of continuous functions with pointwise topology. In the end the thesis shows under which conditions the space of continuous functions satisfies the definition of the respective notions related to compactness. Keywords: spaces of continuous functions; pointwise convergence; compactness; angelicity
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Radon-Nikodým compact spaces
Cepák, Jiří ; Holický, Petr (referee) ; Spurný, Jiří (advisor)
In the present work we study Radon-Nikodým 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.
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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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Open problems in Continuum thory
Seifrt, Jan ; Spurný, Jiří (referee) ; Pyrih, Pavel (advisor)
ISYixev prace: Otevfene problemy teorie konlinm Autor: Jan Seifrt Katcdra (ustav): Katedra matematicke analyzy Vedouci ba.kaliirske prace: Doc. RNDr. Pavel Pyrih, CSc. e-mail vedouciho: poliodai6.gniail.com Anstrakt: PiYdlo/ena pracc sc /a))yva v/tahrin iiia^iicli('l\ycb a koiH-rnr pn- riodickych bodu v jislycli koiiipaktiiu-li .souvislych mno/iuach. Teziste prace s])ociva v in)di-()l)in''in ro/boru dvou ])ul)liku\>uiycli vyslcdkii (motor a null- comb). Fungovam Irclito ])fikladu JL- /achycono na fade pomocnydi obraxkii. Pia(.:c obsabnjc polrubnc definicr a /aktadni 1 vr/cm' Inv, dukaxu. V praci JHOU dale doka'/ana i dalsi tvr/oni / dam'1 problcmatiky. vii shiva: dcndril., ill]I' vla.st.imsr, ina^iiutickr body a mill-comb Tillc: Open problems in Continuum thmry Author: Jan Scifrt Department.; Department of Matheinal ical Analysis Supervisor: Doc. HNDr. Pavel Pyrili, CSc. Supervisor's e-mail address: Abstract: In t lit1 present work \ve study the relation between non-wandering ami eventually-periodic- points in certain compact conned ed sets. The goal of the work consists of detailed study of two published results (engine and null-comb). How these examples work is demonstrated by a. sequence of fi- gures. The work contain all needed definitions a.nd lacts wit.lumt proofs. In the work are proved some other...
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Metric spaces of Hölder mappings
Novotný, Vojtěch ; Spurný, Jiří (referee) ; Hušek, Miroslav (advisor)
Nazcv prace: Metricke prostory hoklerovskych funkci Autor: Vojtech .\ovotny Katcdra: Kiitcrlra matenmticke analyzy Vedouci bakalafske pracc: Prof. liNDr. Miruslav lln.sek, DrSc. E-niail vedouciho: mhusek@karlin.mff.cuni.cz Abstrakt: Ukolem bakalafske prace je definovat prostory a-holderovskych zobrazeni s pfirozenou pseudonormou a studovat jejich vlastuosti. Nesepa- rahilitu, uplnost ci v/tahy jednotlivyoh prostoru vysetfujeme nejdfivc na realnych holdorovskycli fuukcich defiuovaiiych na [0, 1], po/xlcji na jinych omezeuych u/.avfeiiych, ...'li i otevfenych intervalech a nakonec na cole reahu'1 pfiince. V /avererne ka.pitole ziskanc vyslcdky zobecnujeine ]>ro ho'ldcrovska /obrazcni v obecnejsich nietriekyrh prostorech. Klicova slova: holdcrovske zobrazeni, ... zobrazoni. niotricky prostor Title: Metric spaces of Holder mappings Author: Vojtech .\ovotny Department: Department of Mathematical Analysis Supervisor: Prof. KA'Dr. Miroslav Ilusek, DrSc. Supervisor's e-mail address: mhusek@karlin.mff.cuni.cz Abstract: The task of the bachelor work is to define spaces of o-Holder map- pings with their natural pscudononn and to study their properties. First we investigate nonseparability. completeness and relations between particular spaces of real Holder functions defined on [0. l], later...
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Toplogical properties of compact convex sets
Kačena, Miroslav ; Lukeš, Jaroslav (referee) ; Spurný, Jiří (advisor)
The first part of the thesis presents the basics of Choquet theory of function spaces needed in the next part. Text deals mainly with general function spaces, the special case of compact convex sets is considered only marginally. The main object of this investigation is an equivalence between simpliciality and some interpolation properties of a function space. The second part is engaged in research on products of function spaces. Various products are defined, the most treated being the multiaffine product. The introductory section focuses just on the connections and differences between these products. The primary goal of the work is a generalization of known results for products of compact convex sets to the context of function spaces. First, extremal sets are examined, the main result is the representation of Choquet boundary of a product space as the product of Choquet boundaries of original spaces. Simplicial spaces are studied next. It is shown, that a product of simplicial spaces is simplicial and in that case established definitions of a product space coincide for affine functions. Finally, maximal measures are investigated.
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