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Ideals of Banach Spaces
Smetana, Ondřej ; Cúth, Marek (advisor) ; Rmoutil, Martin (referee)
We study a particular class of subspaces of Banach spaces called ideals. We show that the notions of ideals, locally complemented subspaces, and the existence of a Hahn-Banach extension operator coincide. We introduce and develop the method of suitable models, a set-theoretic approach which enables us to write technical proofs in simpler terms. We use the method to prove the existence of an almost isometric ideal. We present applications of almost isometric ideals and the method to the strong and local diameter two properties, and the Daugavet property.
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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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Convergence in Banach Spaces
Silber, Zdeněk ; Kalenda, Ondřej (advisor) ; Plebanek, Grzegorz (referee) ; Cúth, Marek (referee)
The thesis consists of three articles. The common theme of the first two articles is the possibility of iterating weak∗ derived sets in dual Banach spaces. In the first article we prove that in the dual of any non-reflexive Banach space we can always find a convex set of order n for any n ∈ N, and a convex set of order ω +1. This result extends Ostrovskii's characterization of reflexive spaces as those spaces for which weak∗ derived sets coincide with weak∗ closures for convex sets. In the second article we prove an iterated version of another result of Ostrovskii, that a dual to a Banach space X contains a subspace whose weak∗ derived set is proper and norm dense, if and only if X is non-quasi-reflexive and contains an infinite-dimensional subspace with separable dual. In the third article we study quantitative results concerning ξ-Banach-Saks sets and weak ξ-Banach-Saks sets. We provide quantitative analogues to characterizations of weak ξ-Banach-Saks sets using ℓξ+1 1 spreading models and a quantitative version of the relation of ξ-Banach-Saks sets, weak ξ-Banach-Saks sets, norm compactness and weak compactness. We use these results to define a new measure of weak non-compactness and finally give some relevant examples. 1
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Proofs of Tychonoff's Theorem
Dvořáková, Johana ; Cúth, Marek (advisor) ; Holický, Petr (referee)
This bachelor thesis is devoted to four different proofs of Tychonoff's Theorem. The first proof is based on definitions of compact topological space and product topology. The second proof is a construction of convergent subnet of an arbitrary net in a product of compact spaces. The third proof uses the fact that topological space is compact if and only if every universal net is convergent. The last proof is based on characterization of compact spaces using systems of closed subsets with the finite intersection property. 1
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Ulam's problem
Kučerová, Tereza ; Cúth, Marek (advisor) ; Kalenda, Ondřej (referee)
In this bachelor's thesis we deal with Ulam's problem. In the first chapter, we introduce the basic definitions and the axiomatic theory of ZF extended by the Axiom of Choice; we also formulate and prove the Lemma that will be used for the proofs in the second and third chapters. In the second and third chapters, we prove that, assuming the Continuum Hypothesis holds, the Ulam's problem has a positive solution, and assuming the Full Measure Extension Axiom holds, the Ulam's problem has a negative solution. We carry out both proofs with a high degree of detail. Finally, in chapter four, we prove that the generalized Ulam's problem for sets with cardinality greater than that of the real numbers always has a negative solution. 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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Lipschitz Free Spaces and Subsets of Finite-Dimensional Spaces
Bíma, Jan ; Cúth, Marek (advisor) ; Doucha, Michal (referee)
The present thesis is devoted to the geometry of Lipschitz free p-spaces Fp(M) over subsets of finite-dimensional vector spaces, where 0 < p ≤ 1. We solve an open problem and show that if M is an infinite subset of Rd endowed with the H¨older distorted metric | · |α , where 0 < α < 1, then Fp(M, | · |α ) ≃ ℓp for every 0 < p ≤ 1. Moreover, we tackle a question due to Albiac et al. and expound the role of p, d for the Lipschitz constant of a locally coordinatewise affine retraction from (K, | · |1), where K = ⋃︁ Q∈R Q is a union of a collection ∅ ̸= R ⊆ {Rw + R[0, 1]d : w ∈ Zd } of cubes in Rd with side length R > 0, into the Lipschitz free p-space Fp(V, | · |1) over their vertices. The last chapter is then dedicated to the Lipschitz extension problem Lip0(N, Z) → Lip0(M, Z), where N is a doubling subspace of a metric space M and Z is a p-Banach space. As it turns out, the problem can equivalently be stated in terms of a projective relation between the Lipschitz free p-spaces Fp(N) and Fp(M). 1
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Lipschitz free p-spaces
Raunig, Tomáš ; Cúth, Marek (advisor) ; Spurný, Jiří (referee)
This thesis deals with a class of p-Banach spaces known as Lipschitz free p-spaces, where 0 < p ≤ 1. In the first part we describe their construction in detail and give proofs of their basic properties. Using these properties we then characterize the spaces. In the second part we derive a formula, which can under certain circumstances be used to calculate the p-norm on these spaces, and describe an algorithm which calculates the p-norm on finite-dimensional Lipschitz free p-spaces. 1
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Basic properties of p-Banach spaces
Kubíček, David ; Cúth, Marek (advisor) ; Johanis, Michal (referee)
In this thesis we recall the notion of a quasi-norm and a p-norm. We mention the Aoki-Rolewicz theorem which connects these two notions. We deal with generalizations of selected results from Banach spaces to p-Banach spaces for 0 < p ≤ 1. We study the class of Lp(µ) spaces. Namely, we explore their basic properties, their dual spaces and nonlinear structure. 1
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Separable reduction theorems in functional analysis
Cúth, Marek ; Kalenda, Ondřej (advisor) ; Holický, Petr (referee)
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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