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Riemann type integral in Banach spaces
Mrhal, Filip ; Lukeš, Jaroslav (advisor) ; Zajíček, Luděk (referee)
Title: Riemann type integral in Banach spaces Author: Filip Mrhal Department: Department of Mathematical Analysis Supervisor: Prof. RNDr. Jaroslav Lukeš, DrSc., Department of Mathematical Analysis Abstract: In this thesis we study some differences in the behaviours of the Ri- emann integral when integrating functions from any compact subinterval of real numbers to real numbers or to any Banach space. Especially, we outline that the Lebesgue theorem is no longer valid in relationship to functions with images in some Banach spaces. We show that for some well-known Banach spaces using counterexamples. Keywords: Riemann integral, Banach space, Lebesgue theorem 1
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The James theorem and the boundary problem
Lechner, Jindřich ; Spurný, Jiří (advisor) ; Kurka, Ondřej (referee)
Let G be a subset of the dual of a real Banach space X and F ⊂ G. Then F is a James boundary of G if each w∗ -continuous linear functional on X attains its supremum over G on an element of the set F. We ask whether a norm bounded subset of X which is countably compact for the topology generated by F is ne- cessary sequentially compact for the topology generated by G. The main content of our work is a positive solution to this problem. As a corollary we obtain James characterization of weakly compact subsets of a real Banach space. Due to the Eberlein-Šmuljan theorem a positive solution to the so called boundary problem is shown as a special case of the affirmative answer to the question raised above. The question is further discussed for a case of Banach spaces defined over the complex field. In this case we cannot use the old definition of the James boun- dary but by a "natural" way it is possible to redefine the term James boundary and then we are able to answer our question positively again. 1
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Riemann type integral in Banach spaces
Mrhal, Filip ; Lukeš, Jaroslav (advisor) ; Zajíček, Luděk (referee)
Title: Riemann type integral in Banach spaces Author: Filip Mrhal Department: Department of Mathematical Analysis Supervisor: Prof. RNDr. Jaroslav Lukeš, DrSc., Department of Mathematical Analysis Abstract: In this thesis we study some differences in the behaviours of the Ri- emann integral when integrating functions from any compact subinterval of real numbers to real numbers or to any Banach space. Especially, we outline that the Lebesgue theorem is no longer valid in relationship to functions with images in some Banach spaces. We show that for some well-known Banach spaces using counterexamples. Keywords: Riemann integral, Banach space, Lebesgue theorem 1
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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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Microscopic sets and drops in Banach spaces
Pospíšil, Marek ; Lukeš, Jaroslav (advisor) ; Zelený, Miroslav (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 Ekeland's variational principle and its equivalence with the the Daneš's drop theorem, the Brézis-Browder's theorem, the Phelps' lemma and the Caristi-Kirks's theorem. Furthermore, we discuss its relation to the Bishop-Phelps' 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 certain sets in reflexive spaces equivalent.
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The James theorem and the boundary problem
Lechner, Jindřich ; Spurný, Jiří (advisor) ; Kurka, Ondřej (referee)
Let G be a subset of the dual of a real Banach space X and F ⊂ G. Then F is a James boundary of G if each w∗ -continuous linear functional on X attains its supremum over G on an element of the set F. We ask whether a norm bounded subset of X which is countably compact for the topology generated by F is ne- cessary sequentially compact for the topology generated by G. The main content of our work is a positive solution to this problem. As a corollary we obtain James characterization of weakly compact subsets of a real Banach space. Due to the Eberlein-Šmuljan theorem a positive solution to the so called boundary problem is shown as a special case of the affirmative answer to the question raised above. The question is further discussed for a case of Banach spaces defined over the complex field. In this case we cannot use the old definition of the James boun- dary but by a "natural" way it is possible to redefine the term James boundary and then we are able to answer our question positively again. 1
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