The universal renorming (Q466141)

The author introduces the universal renorming technique. It gives a formal way to determine any equivalent renorming of a real normed space. Using this tool, the author proves that every real normed space with property \(\alpha\) admits an equivalent renorming so that the set of non norm-attaining functionals is nowhere dense. In particular, this applies to the class of weakly compactly generated spaces. This is achieved by studying the algebraic and geometrical properties of the universal renorming and their relationship with strongly exposed points and smooth faces of the unit ball of the space.