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Topological and descriptive methods in the theory of function and Banach spaces
Kačena, Miroslav
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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Topological and descriptive methods in the theory of function and Banach spaces
Kačena, Miroslav
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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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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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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