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Banach-Mazur distance between Banach spaces of continuous functions
Havelka, Jonáš ; Cúth, Marek (advisor) ; Rondoš, Jakub (referee)
In this bachelor thesis, we examine estimates of a Banach-Mazur distance between spaces C(ω) and C(ω · k) for various k ∈ N. With a properly chosen form of mapping, we constructively show the best currently known upper bounds of this distance for all k. Besides the known bounds d(C(ω), C(ω · k)) ≤ 2 + √ 5 for any k > 3, and d(C(ω), C(ω · 2)) ≤ 3, we are going to find, for k = 3, the better bound, namely d(C(ω), C(ω · 3)) ≤ 4 instead of already known 2 + √ 5. 1
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Vector-valued integral representation
Rondoš, Jakub ; Spurný, Jiří (advisor) ; Galego, Eloi Medina (referee) ; Cúth, Marek (referee)
The thesis consists of seven research papers. The first two papers study the properties of fragmented convex functions, mainly the so-called maximum principle. The first paper deals with convex functions defined on compact convex subsets of locally convex spaces, the second one with the abstract convex functions defined on general compact Hausdorff spaces. The next four papers present results in the spirit of the well-known Banach-Stone theorem in the area of subspaces of continuous functions. The first of those four papers deals with the spaces of affine continuous complex functions on compact convex sets. The second paper extends the results of the first one to the context of general subspaces of continuous functions defined on locally compact spaces. The other two papers further extend the previous results to the case of Banach space-valued and Banach lattice-valued functions, respectively. The last paper is devoted to the study of the Banach-Mazur distance between subspaces of vector-valued continuous functions that have scattered boundaries. 1
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Disintegration of category in the sense of the Kuratowski-Ulam theorem
Rondoš, Jakub ; Spurný, Jiří (advisor) ; Holický, Petr (referee)
The Fubini theorem and the Kuratowski-Ulam theorem show similarities be- tween the concepts meager set in Polish space and set of zero measure in standard Borel space. Those theorems can be generalized. The Fubini theorem is generali- zed by the Measure disintegration theorem and the Kuratowski-Ulam theorem is generalized by the Category disintegration theorem. The claims of disintegration theorems are analogous and show even more the similarities between meager sets and sets of zero measure. The main aim of this thesis is to prove disintegration theorems and show how Fubini and Kuratowski-Ulam theorems follow. 1
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