 

Quantitative weak compactness
Rolínek, Michal ; Spurný, Jiří (advisor) ; Kalenda, Ondřej (referee)
In this thesis we study quantitative weak compactness in spaces (C(K), τp) and later in Banach spaces. In the first chapter we introduce several quantities, which in different manners measure τpnoncompactness of a given uniformly bounded set H ⊂ RK . We apply the results in Banach spaces in chapter 2, where we prove (among others) a quantitative version of the EberleinSmulyan theorem. In the third chapter we focus on convex closures and how they affect measures of noncompactness. We prove a quantitative version of the KreinSmulyan theorem. The first three chapters show that measuring noncompactness is intimately related to measuring distances from function spaces. We follow this idea in chapters 4 and 5, where we measure distances from Baire one functions first in RK and later also in Banach spaces. 1


Properties of deltamatroids
Šíma, Lucien ; Kazda, Alexandr (advisor) ; Rolínek, Michal (referee)
We investigate deltamatroids which are formed by families of subsets of a finite ground set such that the exchange axiom is satisfied. We deal with some natural classes of deltamatroids. The main result of this thesis establishes sev eral relations between even, linear, and matchingrealizable deltamatroids. Fol lowing up on the ideas due to Geelena, Iwatab, and Murota [2003], and apply ing the properties of field extensions from algebra, we prove that the class of strictly matchingrealizable deltamatroids, the subclass of matchingrealizable deltamatroids, is included in the class of linear deltamatroids. We also show that not every linear deltamatroid is matchingrealizable by giving a skewsymmetric matrix representation to the non matchingrealizable deltamatroid constructed by Kazda, Kolmogorov, and Rol'ınek [2019].


Quantitative weak compactness
Rolínek, Michal ; Spurný, Jiří (advisor) ; Kalenda, Ondřej (referee)
In this thesis we study quantitative weak compactness in spaces (C(K), τp) and later in Banach spaces. In the first chapter we introduce several quantities, which in different manners measure τpnoncompactness of a given uniformly bounded set H ⊂ RK . We apply the results in Banach spaces in chapter 2, where we prove (among others) a quantitative version of the EberleinSmulyan theorem. In the third chapter we focus on convex closures and how they affect measures of noncompactness. We prove a quantitative version of the KreinSmulyan theorem. The first three chapters show that measuring noncompactness is intimately related to measuring distances from function spaces. We follow this idea in chapters 4 and 5, where we measure distances from Baire one functions first in RK and later also in Banach spaces. 1

 