This monograph presents an up-to-date panorama of the different
techniques and results in the large field of renorming in Banach spaces
and its applications. The reader will find a self-contained exposition
of the basics on convexity and differentiability, the classical results
in building equivalent norms with useful properties, and the evolution
of the subject from its origin to the present days. Emphasis is done on
the main ideas and their connections.
The book covers several goals. First, a substantial part of it can be
used as a text for graduate and other advanced courses in the geometry
of Banach spaces, presenting results together with proofs, remarks and
developments in a structured form. Second, a large collection of recent
contributions shows the actual landscape of the field, helping the
reader to access the vast existing literature, with hints of proofs and
relationships among the different subtopics. Third, it can be used as a
reference thanks to comprehensive lists and detailed indices that may
lead to expected or unexpected information.
Both specialists and newcomers to the field will find this book
appealing, since its content is presented in such a way that
ready-to-use results may be accessed without going into the details.
This flexible approach, from the in-depth reading of a proof to the
search for a useful result, together with the fact that recent results
are collected here for the first time in book form, extends throughout
the book. Open problems and discussions are included, encouraging the
advancement of this active area of research.
