Some Open Problems

Abstract

We have extensively considered here the use of Stone's theorem on the paracompactness of metric spaces in order to build up new techniques to construct an equivalent locally uniformly rotund norm on a given normed space X. The discreetness of the basis for the metric topologies gives us the necessary rigidity condition that appears in all the known cases of existence of such a renorming property [Hay99, MOTV06]. Our approximation process is based on co-σ-continuous maps using that they have separable fibers, see Sect. 2.2. We present now some problems that remain open in this area. Some of them are classical and have been asked by different authors in conferences, papers and books. Others have been presented in schools, workshops, conferences and recent papers on the matter and up to our knowledge they remain open. The rest appear here for the first time. We apologize for any fault assigning authorship to a given question. Rather than to formulate precise evaluation for the first time the problems were proposed, our aim is to provide good questions for young mathematicians entering in the field, we think they deserve all our attention to complete the state of the art in renorming theory.