The field of renorming theory in Banach spaces focuses on adjusting the norm while preserving topological properties. This theory is instrumental in unveiling the underlying geometric structures within Banach spaces and generating valuable mathematical insights. Traditionally, Banach spaces have been categorized based on their topological and geometrical properties. In this study, we narrow our focus to investigate the relationship between three fundamental aspects: the norm's geometric properties, smoothness, and separability of a space. It was previously established that every separable Banach space could be renormed with locally uniformly rotund (LUR) norms that are also Gâteaux differentiable. Similarly, separable Banach spaces with separable duals could be renormed with LUR norms that are also Fréchet differentiable. However, we tackle two open problems that question the existence of renormings for spaces possessing these topological properties but with weaker geometric norm properties. Our objectives are to prove the existence of norms that are rotund, Gâteaux differentiable, and not midpoint locally uniformly rotund (MLUR) for separable Banach spaces. Additionally, we aim to establish the existence of renormings that are weakly uniformly rotund, Fréchet differentiable, and not MLUR. This research delves into the nuanced relationship between topological and geometric properties of Banach spaces, contributing to the broader understanding of these spaces in functional analysis.
Il campo della teoria dei rinormamenti negli spazi di Banach si concentra sulla modifica della norma preservando le proprietà topologiche. Questa teoria è fondamentale per studiare le strutture geometriche degli spazi di Banach attraverso cui si possono classificare in maniera efficace. In questa tesi, restringiamo la nostra attenzione sulla relazione tra tre aspetti fondamentali: le proprietà geometriche della norma, la regolarità e la separabilità di uno spazio. In precedenza, è stato dimostrato che ogni spazio di Banach separabile poteva essere ri-normato con norme localmente uniformemente convesse (LUR) che sono anche Gâteaux differenziabili. Allo stesso modo, gli spazi di Banach separabili con duale separabile potevano essere ri-normati con norme localmente uniformemente convesse che sono anche Fréchet differenziabili. I nostri obiettivi sono dunque dimostrare l'esistenza di norme che sono strettamente convesse, Gâteaux differenziabili e non MLUR per spazi di Banach separabili. Inoltre, si proverà a stabilire l'esistenza di rinormamenti che sono debolmente uniformemente convessi, Fréchet differenziabili e non MLUR in spazi di Banach con duale separabile. Questa ricerca approfondisce la complessa relazione tra le proprietà topologiche e geometriche degli spazi di Banach, contribuendo alla comprensione più ampia di questi spazi nell'ambito dell'analisi funzionale.
On the existence of rotund Gâteaux smooth norms which are not midpoint locally uniformly rotund
Preti, Alessandro
2022/2023
Abstract
The field of renorming theory in Banach spaces focuses on adjusting the norm while preserving topological properties. This theory is instrumental in unveiling the underlying geometric structures within Banach spaces and generating valuable mathematical insights. Traditionally, Banach spaces have been categorized based on their topological and geometrical properties. In this study, we narrow our focus to investigate the relationship between three fundamental aspects: the norm's geometric properties, smoothness, and separability of a space. It was previously established that every separable Banach space could be renormed with locally uniformly rotund (LUR) norms that are also Gâteaux differentiable. Similarly, separable Banach spaces with separable duals could be renormed with LUR norms that are also Fréchet differentiable. However, we tackle two open problems that question the existence of renormings for spaces possessing these topological properties but with weaker geometric norm properties. Our objectives are to prove the existence of norms that are rotund, Gâteaux differentiable, and not midpoint locally uniformly rotund (MLUR) for separable Banach spaces. Additionally, we aim to establish the existence of renormings that are weakly uniformly rotund, Fréchet differentiable, and not MLUR. This research delves into the nuanced relationship between topological and geometric properties of Banach spaces, contributing to the broader understanding of these spaces in functional analysis.File | Dimensione | Formato | |
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https://hdl.handle.net/10589/214913