Geometrical structure in diameter 2 Banach spaces
Date
2015-07-02
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Abstract
Banachi ruumide geomeetrias mängib olulist rolli tema ühikkera struktuur. Varasemalt on põhjalikult uuritud erinevaid Banachi ruumide omadusi ja mõisteid, mille üheks tunnuseks on ühikkera kui tahes väikese läbimõõduga viilude olemasolu, näiteks Radon–Nikodými omadus, hammaspunktid või tugevalt eksponeeritud punktid. Alles suhteliselt hiljuti on eraldi hakatud uurima teises suunas äärmuslikke omadusi, mille puhul ühikkera iga viilu diameeter on kaks, see omadus on näiteks klassikalistel Banachi ruumidel c0, ℓ∞, C[0,1] ja L1[0,1]. Neis ruumides on isegi ühikkera ja nõrgalt lahtise osahulga mittetühja lõike (erijuhul viilu) diameeter kaks. Viimast omadust on hakatud nimetama diameeter-2 omaduseks. Käesolevas väitekirjas käsitletakse diameeter-2 omaduse peamisi uurimissuundi: selgitatakse diameeter-2 omadusega sarnaste omaduste omavaheline vahekord, esitletakse uusi näiteid, kirjeldatakse diameeter-2 omadusega Banachi ruumide kaasruume, selgitatakse diameeter-2 omaduste üle kandumist alam- ja ülemruumidele ning ruumide summale eri normide korral, selgitatakse operaatorite ruumi struktuuri seost vastava siht- ja lähteruumi struktuuriga.
In the geometry of Banach spaces structure of the unit ball plays an important role. Previously, various properties and notions describing Banach spaces whose unit ball has slices with arbitrarily small diameter have been extensively studied, for example, the Radon–Nikodým property, denting points or strongly exposed points. Lately, the extreme opposite property – Banach spaces with all slices of the unit ball having diameter equal to 2 – has gotten some extra attention. For example, the classical Banach spaces c0, ℓ∞, C[0,1], and L1[0,1] all have this property. These spaces even have the property that every nonempty relatively weakly open subset (in particular a slice) of their unit ball has diameter equal to 2. The latter property is called the diameter 2 property. In the current thesis, diameter 2 property and similar properties are studied, dual spaces of diameter 2 spaces are characterized in terms of octahedrality, stability results of diameter 2 properties are presented, and relations between diameter 2 structure of spaces of operators and corresponding Banach spaces are investigated.
In the geometry of Banach spaces structure of the unit ball plays an important role. Previously, various properties and notions describing Banach spaces whose unit ball has slices with arbitrarily small diameter have been extensively studied, for example, the Radon–Nikodým property, denting points or strongly exposed points. Lately, the extreme opposite property – Banach spaces with all slices of the unit ball having diameter equal to 2 – has gotten some extra attention. For example, the classical Banach spaces c0, ℓ∞, C[0,1], and L1[0,1] all have this property. These spaces even have the property that every nonempty relatively weakly open subset (in particular a slice) of their unit ball has diameter equal to 2. The latter property is called the diameter 2 property. In the current thesis, diameter 2 property and similar properties are studied, dual spaces of diameter 2 spaces are characterized in terms of octahedrality, stability results of diameter 2 properties are presented, and relations between diameter 2 structure of spaces of operators and corresponding Banach spaces are investigated.
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Banachi ruumid, Banach spaces