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Quantitative weak compactness
Rolínek, Michal ; Spurný, Jiří (advisor) ; Kalenda, Ondřej (referee)
In this thesis we study quantitative weak compactness in spaces (C(K), τp) and later in Banach spaces. In the first chapter we introduce several quantities, which in different manners measure τp-noncompactness of a given uniformly bounded set H ⊂ RK . We apply the results in Banach spaces in chapter 2, where we prove (among others) a quantitative version of the Eberlein-Smulyan theorem. In the third chapter we focus on convex closures and how they affect measures of noncompactness. We prove a quantitative version of the Krein-Smulyan theorem. The first three chapters show that measuring noncompactness is intimately related to measuring distances from function spaces. We follow this idea in chapters 4 and 5, where we measure distances from Baire one functions first in RK and later also in Banach spaces. 1
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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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Separable reduction theorems in functional analysis
Cúth, Marek ; Holický, Petr (referee) ; Kalenda, Ondřej (advisor)
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.
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