Transfinite geometric properties of the unit ball in Banach spaces
Date
2024-06-20
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Abstract
Banachi ruumide geomeetrias mängib olulist rolli tema ühikkera struktuur. Mis tahes ühikkera alamhulga diameeter on ülimalt kaks. Erilist tähelepanu on pälvinud ühikkera viilud, mis on ühikkera teatud alamhulgad. Öeldakse, et Banachi ruumil on lokaalne diameeter-2 omadus, kui tema ühikkera iga viilu diameeter on kaks. Selline ekstremaalne nähtus leiab aset vaid lõpmatumõõtmelistes Banachi ruumides. Diameeter-2 omaduste ja nendega seotud mõistete süstemaatilise uurimise käivitasid 2013. aastal T. A. Abrahamsen, V. Lima ja O. Nygaard. Seejärel on kirjanduses uuritud erinevaid diameeter-2 omaduse tugevdusi, nt peaaegu ruudu omadust ja oktaeedrilised norme.
Kõigi nende ülaltoodud mõistete ühine omadus on see, et need on nii öelda lõplikult määratletud; see tähendab, et definitsioonid kasutavad seda, et mis tahes lõpliku arvu elementide jaoks Banachi ruumis või tema kaasruumis leidub mingi eriline element selles samas ruumis või tema kaasruumis, mis annab vastava omaduse. Sellised geomeetrilised omadused üldistas J. D. Hardtke testperekonna nime alla 2020. aastal. Üllataval kombel on paljudel klassikalistel Banachi ruumidel isegi diameeter-2 omaduste või nendega seotud omaduste transfiniitne analoog.
Käesoleva väitekirja põhieesmärk on süstemaatiliselt uurida klassikaliste diameeter-2 omaduste, peaaegu ruudu omaduse ja oktaeedriliste normide transfiniitseid analooge. Üldiselt käituvad transfiniitsed analoogid olemasolevatest diameeter-2 omadustest erinevalt ja on tehniliselt keerulisemad. Seega annab see suund uusi viljakaid tulemusi ja näiteid, mis oluliselt täiendavad olemasolevat diameeter-2 omadusega ruumide ja nendega seotud mõistete teooriat.
The structure of the unit ball plays an important role in understanding the geometry of Banach spaces. At most, the maximal diameter of a subset of the unit ball in a Banach space can be two. Special interest has been turned to determining the diameter of slices, which are formed by intersecting of the unit ball with half-spaces. If every slice of the unit ball of a Banach space has diameter two, then the space is said to have the local diameter two property. This extreme geometric phenomenon can only happen in infinite-dimensional Banach spaces. A systematic treatment of diameter two properties and their relatives was started by T. A. Abrahamsen, V. Lima, and O. Nygaard in 2013. From there on, various strengthenings of diameter two properties and their relatives have emerged, e.g., almost square spaces and octahedral norms. A common property for all of these notions above is that they are finitely defined; that is, the definitions require that for every finite number of elements in a Banach space or its dual, there is some special element in the space or the dual. Such geometric properties were formalized by J. D. Hardtke as test families in 2020. Surprisingly, many classical Banach spaces enjoy even a transfinite analogue of the diameter two properties or their relatives. The main aim of the thesis is to systematically study transfinite extensions of the classical diameter two properties, almost squareness, and octahedral norms. The transfinite case generally exhibits significantly distinct behavior and is technically more complex. Hence, it gives new fruitful results and examples that substantially complement the existing theory of spaces with the diameter two properties and their relatives.
The structure of the unit ball plays an important role in understanding the geometry of Banach spaces. At most, the maximal diameter of a subset of the unit ball in a Banach space can be two. Special interest has been turned to determining the diameter of slices, which are formed by intersecting of the unit ball with half-spaces. If every slice of the unit ball of a Banach space has diameter two, then the space is said to have the local diameter two property. This extreme geometric phenomenon can only happen in infinite-dimensional Banach spaces. A systematic treatment of diameter two properties and their relatives was started by T. A. Abrahamsen, V. Lima, and O. Nygaard in 2013. From there on, various strengthenings of diameter two properties and their relatives have emerged, e.g., almost square spaces and octahedral norms. A common property for all of these notions above is that they are finitely defined; that is, the definitions require that for every finite number of elements in a Banach space or its dual, there is some special element in the space or the dual. Such geometric properties were formalized by J. D. Hardtke as test families in 2020. Surprisingly, many classical Banach spaces enjoy even a transfinite analogue of the diameter two properties or their relatives. The main aim of the thesis is to systematically study transfinite extensions of the classical diameter two properties, almost squareness, and octahedral norms. The transfinite case generally exhibits significantly distinct behavior and is technically more complex. Hence, it gives new fruitful results and examples that substantially complement the existing theory of spaces with the diameter two properties and their relatives.