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LUR renormings through Deville’s master lemma. (English) Zbl 1192.46008

Much effort has been spent finding necessary and sufficient conditions on a given Banach space for the existence of an equivalent locally uniformly rotund norm, preferably one which is also lower semicontinuous with respect to the weak topology induced by a suitable norming subspace of the dual. The main purpose of this paper is to give more geometric proofs of some important (albeit technical) results in this area. In particular, they do not rely on the classical fact that all metric spaces are paracompact. The proofs of sufficiency contain some quantitative improvements over earlier arguments. The final section reframes these results in the context of topological networks.

MSC:

46B03 Isomorphic theory (including renorming) of Banach spaces
46B26 Nonseparable Banach spaces
46B20 Geometry and structure of normed linear spaces
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