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Descriptive and topological aspects of Banach space theory
Kurka, Ondřej ; Holický, Petr (advisor) ; Fabian, Marián (referee) ; Hájek, Petr (referee)
The thesis consists of three papers of the author. In the first paper, it is shown that the sets of Fréchet subdifferentiability of Lipschitz functions on a Banach space X are Borel if and only if X is reflexive. This answers a ques- tion of L. Zajíček. In the second paper, a problem of G. Debs, G. Godefroy and J. Saint Raymond is solved. On every separable non-reflexive Banach space, equivalent strictly convex norms with the set of norm-attaining func- tionals of arbitrarily high Borel class are constructed. In the last paper, binormality, a separation property of the norm and weak topologies of a Ba- nach space, is studied. A result of P. Holický is generalized. It is shown that every Banach space which belongs to a P-class is binormal. It is also shown that the asplundness of a Banach space is equivalent to a related separation property of its dual space. 1
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Isomorphic and isometric classification of spaces of continuous and Baire affine functions
Ludvík, Pavel ; Spurný, Jiří (advisor) ; Kalenda, Ondřej (referee) ; Fabian, Marián (referee)
This thesis consists of five research papers. The first paper: We prove that under certain conditions, the existence of an isomorphism between spaces of continuous affine functions on the compact convex sets imposes home- omorphism between the sets of its extreme points. The second: We investigate a transfer of descriptive properties of elements of biduals of Banach spaces con- strued as functions on dual unit balls. We also prove results on the relation of Baire classes and intrinsic Baire classes of L1-preduals. The third: We identify intrinsic Baire classes of X with the spaces of odd or homogeneous Baire functions on ext BX∗ , provided X is a separable real or complex L1-predual with the set of extreme points of its dual unit ball of type Fσ. We also provide an example of a separable C∗ -algebra such that the second and second intrinsic Baire class of its bidual differ. The fourth: We generalize some of the above mentioned results for real non-separable L1-preduals. The fifth: We compute the distance of a general mapping to the family of mappings of the first resolvable class via the quantity frag and we introduce and investigate a class of mappings of countable oscillation rank.
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Microscopic sets and drops in Banach spaces
Pospíšil, Marek ; Lukeš, Jaroslav (advisor) ; Fabian, Marián (referee)
First we define microscopic sets on the real axis and study their relation to the sets of Hausdorff and Lebesgue measure zero and the sets of first category. In the second part, we prove the Bishop-Phelps' theorem and its equivalence with the Ekeland's variational principle, the Daneš's drop theorem, the Brézis-Browder's theorem and the Caristi-Kirks's theorem. Doing so we define the notion of a drop as the convex hull of a set and a point. In the third part we prove that the drop property equals reflexivity in some sense. A space has the drop property if it is possible to find the drop from the Daneš's theorem even in a more general case than the theorem itself guarantees. Furthermore, we characterize this property using the approximative compactness. Last, we study the microscopic drop property that is more relaxed than the original drop property. We find out that those two notions are for noncompact sets in reflexive spaces equivalent. Powered by TCPDF (www.tcpdf.org)
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Isomorphic and isometric classification of spaces of continuous and Baire affine functions
Ludvík, Pavel ; Spurný, Jiří (advisor) ; Kalenda, Ondřej (referee) ; Fabian, Marián (referee)
This thesis consists of five research papers. The first paper: We prove that under certain conditions, the existence of an isomorphism between spaces of continuous affine functions on the compact convex sets imposes home- omorphism between the sets of its extreme points. The second: We investigate a transfer of descriptive properties of elements of biduals of Banach spaces con- strued as functions on dual unit balls. We also prove results on the relation of Baire classes and intrinsic Baire classes of L1-preduals. The third: We identify intrinsic Baire classes of X with the spaces of odd or homogeneous Baire functions on ext BX∗ , provided X is a separable real or complex L1-predual with the set of extreme points of its dual unit ball of type Fσ. We also provide an example of a separable C∗ -algebra such that the second and second intrinsic Baire class of its bidual differ. The fourth: We generalize some of the above mentioned results for real non-separable L1-preduals. The fifth: We compute the distance of a general mapping to the family of mappings of the first resolvable class via the quantity frag and we introduce and investigate a class of mappings of countable oscillation rank.
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Descriptive and topological aspects of Banach space theory
Kurka, Ondřej ; Holický, Petr (advisor) ; Fabian, Marián (referee) ; Hájek, Petr (referee)
The thesis consists of three papers of the author. In the first paper, it is shown that the sets of Fréchet subdifferentiability of Lipschitz functions on a Banach space X are Borel if and only if X is reflexive. This answers a ques- tion of L. Zajíček. In the second paper, a problem of G. Debs, G. Godefroy and J. Saint Raymond is solved. On every separable non-reflexive Banach space, equivalent strictly convex norms with the set of norm-attaining func- tionals of arbitrarily high Borel class are constructed. In the last paper, binormality, a separation property of the norm and weak topologies of a Ba- nach space, is studied. A result of P. Holický is generalized. It is shown that every Banach space which belongs to a P-class is binormal. It is also shown that the asplundness of a Banach space is equivalent to a related separation property of its dual space. 1
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Separable reductions and rich families in theory of Fréchet subdifferentials
Fabian, Marián
We consider important properties of Fréchet subdifferentials, in particular: the non-emptiness of subdifferentials, the non-zeroness of normal cones, the fuzzy calculus, and the extremal principle; all statements being considered in Fréchet sense. Given a nonseparable Banach space X, we show how the validity of these statements is implied by the validity of them in every separable subspace of X. Such a reasoning is called “separable reduction”. We show that, behind this approach, there is a modern and powerful concept of rich subfamily of the family of all separable subspaces of X.
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