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Abstract In this paper we prove that every infinite-dimensional and separable Banach space ( X , ‖ ⋅ ‖ X ) admits an equivalent norm ‖ ⋅ ‖ X ,… Click to show full abstract
Abstract In this paper we prove that every infinite-dimensional and separable Banach space ( X , ‖ ⋅ ‖ X ) admits an equivalent norm ‖ ⋅ ‖ X , 1 such that ( X , ‖ ⋅ ‖ X , 1 ) has both the Kadec-Klee and the Opial properties. This result also has a quantitative aspect and when combined with the properties of Schauder bases and the Day norm it constitutes a basic tool in the proof of our main theorem: each infinite-dimensional, reflexive and separable Banach space ( X , ‖ ⋅ ‖ X ) has an equivalent norm ‖ ⋅ ‖ 0 such that ( X , ‖ ⋅ ‖ 0 ) is LUR and contains a diametrically complete set with empty interior.
Journal Title: Journal of Functional Analysis
Year Published: 2020
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