Original title: Míry slabé nekompaktnosti v klasických prostorech
Translated title: Measures of weak non-compactness in classical spaces
Authors: Smerčiak, Samuel ; Kalenda, Ondřej (advisor) ; Kurka, Ondřej (referee)
Document type: Master’s theses
Year: 2026
Language: slo
Abstract: [eng] [cze]
In this thesis, we deal with the measures of weak non-compactness wck, wk, and ω in Banach spaces. In the beginning, we give their definitions and basic properties, while summarizing known results on the equivalence of these measures in some Banach spaces. Next we prove the equality of the measures wck and wk in separable spaces and the equality of the measures wk and ω in the space c0(Γ). We give a brief overview of facts about C(K) spaces with emphasis on the case of a countable compact K, and we subsequently use these to obtain a characterization of the equivalence of the measures wck, wk, and ω in the spaces C(K) = C(K, R) for a metrizable compact K, including the optimality of the equivalence constants. As a consequence, we obtain that the measures wk and ω are not equivalent in the space C([0, 1]), which is the first known example of a classical Banach space that satisfies this. V tejto práci sa zaoberáme mierami slabej nekompaktnosti wck, wk a ω v Banachových priestoroch. Na začiatku uvedieme ich definície a základné vlastnosti, pričom zhrnieme známe výsledky o ekvivalencii týchto mier v niektorých Banachových priestoroch. Ďalej dokazujeme rovnosť mier wck a wk v separabilných priestoroch a rovnosť mier wk a ω v priestore c0(Γ). Uvedieme stručný prehľad faktov o priestoroch C(K) s dôrazom na prípad spočetného kompaktu K a tieto následne použijeme, aby sme získali charakterizáciu ekvivalencie mier wck, wk a ω v priestoroch C(K) = C(K, R) pre metrizovateľný kompakt K vrátane optimality konštánt ekvivalencie. Ako dôsledok dostaneme, že miery wk a ω nie sú ekvivalentné v priestore C([0, 1]), pričom ide o prvý známy príklad klasického Banachovho priestoru, ktorý to spĺňa.
Keywords: measures of weak non-compactness|the space of continuous functions on a compact set|equivalence constants; miery slabej nekompaktnosti|priestor spojitých funkcií na kompakte|konštanty ekvivalencie

Institution: Charles University Faculties (theses) (web)
Document availability information: Available in the Charles University Digital Repository.
Original record: http://hdl.handle.net/20.500.11956/209534

Permalink: http://www.nusl.cz/ntk/nusl-779702


The record appears in these collections:
Universities and colleges > Public universities > Charles University > Charles University Faculties (theses)
Academic theses (ETDs) > Master’s theses