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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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Convex subsets of dual Banach spaces
Silber, Zdeněk ; Kalenda, Ondřej (advisor) ; Spurný, Jiří (referee)
The main topic of this thesis is separation of points and w∗ -derived sets in dual Banach spaces. We show, that in duals of reflexive spaces w∗ -derived set of a convex subset coincides with its w∗ -closure. We also show, that subspace of a dual reflexive space is norming, if and only if it is total. Later we show, that in the dual of every non-reflexive space we can find a convex subset whose w∗ -derived set is not w∗ -closed. Hence, this statement is a characterisation of reflexive spaces. Next we show, that subspaces in duals of quasi-reflexive spaces are norming, if and only if they are total. Later we show, that in the dual of every non-quasi-reflexive space we can find a subspace which is total but not norming; thus, the previous statement is a characterisation of quasi-reflexive spaces. We also show, that for absolutely convex subsets of duals of quasi-reflexive spaces w∗ -derived set coincides with w∗ -closure. In the last section we define w∗ -derived sets of higher orders and show, that in the dual of every non-quasi-reflexive separable Banach space there exist subspaces of order of each countable non-limit ordinal and no other. 1
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Impossible sets
Silber, Zdeněk ; Pražák, Dalibor (advisor) ; Zelený, Miroslav (referee)
In this work we de fine Hausdorff measure and dimension, describe the geometrical construction of a Besikovitch set and adapt this approach to construct a Kakeya set. We also describe another construction of a Besicovitch set using the properties of projections of irregular sets. Finally we present other examples of "impossible sets". Powered by TCPDF (www.tcpdf.org)
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