LOCALIZATION TECHNIQUES FOR RENORMING

Renorming theory involves finding isomorphisms in order to improve the norm of a normed space X. This means to make the geometrical and topological properties of the unit ball of the given normed space as close as possible to those of the unit ball of a Hilbert space. In this work we study different types of geometrical properties. In 1989 Hansell introduced the notion of descriptive topological space: we do not state this definition here, since it is rather technical. Hansell pointed out the role played by the existence of sigma-isolated networks in these spaces. They can replace sigma-discrete topological bases which turn out to be exclusive of metrizable space after the Bing--Nagata--Smirnov theorem. So Hansell proved that a Banach space is descriptive with respect to the weak-topology if, and only if, the norm topology has a sigma-isolated network with respect to the weak-topology. Hansell also proved that if a Banach space has a Kadec norm, then it is descriptive with respect to the weak-topology The main problem in Kadec renorming theory is whether it is possible to prove the converse of the previous theorem: in fact no examples are known of descriptive Banach spaces, with respect to the weak-topology, that do not admit an equivalent Kadec norm. In this work we prove the following theorem: X is a descriptive Banach space with respect to the weak-topology if, and only if, there exists an equivalent weak-lower semicontinuous and weak-Kadec quasinorm q(.), i.e. a quasinorm such that the weak and the norm topologies coincide on the set {x in X s.t. q(x)=1} and a||x||< q(x)<\b||x|| holds for some positive constants a and b. In the second part of this dissertation we state some results on rotund renormings. In this work we will give a characterization of rotund renorming in term of the G$_\delta$-diagonal property: X admits an equivalent, weak-lower semicontinuous and rotund norm if, and only if, X admits an equivalent, weak-lower semicontinuous norm ||.|| such that the set {x in X s.t. ||x||=1} has a G$_\delta$-diagonal with slices. We also prove some transference results. In the third part of the dissertation we begin a study of uniformly rotund renorming theory.