Charakterisierungen schwacher Kompaktheit in Dualräumen

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Titel: Charakterisierungen schwacher Kompaktheit in Dualräumen
Sonstige Titel: Characterizations of weak compactness in dual spaces
Autor(en): Möller, Christian
Erstgutachter: Prof. Dr. Peter Meyer-Nieberg
Zweitgutachter: Prof.Dr. Egon Scheffold
Zusammenfassung: In this thesis we present an extensive characterization of weak* sequentially precompact subsets of the dual of a sequentially order complete M-space with an order unit. This central part of the thesis generalizes results due to H.H. Schaefer and X.D. Zhang showing that small weak* compact subsets of the dual of a space of bounded measurable real-valued functions (continuous real-valued functions on a compact quasi-Stonian space) are weakly compact. Moreover, while the proofs of Schaefer and Zhang use measure theoretical arguments, the arguments presented here are purely elementary and are based on the well-known result, that the space l1 has the Schur property. Finally some applications are given. For example, we investigate compact or sequentially precompact subsets, which consist of order-weakly compact operators, in the space of continuous linear operators defined on a sequentially order complete Riesz space with values in a Banach space provided with the strong operator topology: as an immediate consequence of the results, we can easily deduce extended versions of the Vitali-Hahn-Saks theorem for vector measures. For this we need a generalization of the Yosida-Hewitt decomposition theorem, which is proved here with other techniques like the factorization of an order-weakly compact operator through a Banach lattice with order continuous norm.
URL: https://repositorium.ub.uni-osnabrueck.de/handle/urn:nbn:de:gbv:700-2003091515
Schlagworte: weak compactness; M-space with an order unit; dual space; Schur property; Rosenthal´s Lemma; (order-)weakly compact operator; Vitali-Hahn-Saks theorem
Erscheinungsdatum: 15-Sep-2003
Enthalten in den Sammlungen:FB06 - E-Dissertationen

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