Renormings of Nonseparable Reflexive Banach Spaces and Diametrically Complete Sets with Empty Interior

Abstract

We prove that for each nonseparable and reflexive Banach space (X, • X ) with the nonstrict Opial and Kadec-Klee properties, there exists an equivalent norm• 0 such that the Banach space (X, • 0 ) is LUR and contains a diametrically complete set with empty interior.

Diametrically Complete Sets with Empty Interior and Constant Width Sets with Empty Interior

“…More precisely, in [7] it is proved that such a norm exists for each infinite-dimensional, separable and reflexive Banach space. In [24] we extend this result and prove that for each infinite-dimensional reflexive Banach space (X, • X ) with the nonstrict Opial and the Kadec-Klee properties, there exists an equivalent norm such that X equipped with this norm is LUR and contains a diametrically complete set with empty interior. These additional assumptions on the Banach space (X, • X ) indicate the limit of applications of the Maluta method 0123456789().…”

“…Thus our paper is a complement to the papers [7], [24] and [39]. More precisely, in [7] it is proved that such a norm exists for each infinite-dimensional, separable and reflexive Banach space.…”

“…Throughout this paper all Banach spaces are over the field of reals. We use the notations and notions given in [24]. In the sequel we also use the standard definitions of unconditional series, unconditional basic sequence, Schauder basis and unconditional Schauder basis.…”