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Strictly convex renormings
Author(s) -
Moltó A.,
Orihuela J.,
Troyanski S.,
Zizler V.
Publication year - 2007
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdm040
Subject(s) - mathematics , uncountable set , banach space , normed vector space , strictly convex space , reflexive space , dual polyhedron , pure mathematics , regular polygon , unit sphere , norm (philosophy) , schauder basis , uniformly convex space , pointwise , convex set , locally convex topological vector space , subderivative , characterization (materials science) , class (philosophy) , discrete mathematics , lp space , interpolation space , banach manifold , topological space , functional analysis , mathematical analysis , convex optimization , countable set , computer science , materials science , artificial intelligence , law , chemistry , biochemistry , geometry , political science , gene , nanotechnology
Abstract A normed space X is said to be strictly convex if x = y whenever ‖( x + y )/2‖ = ‖ x ‖ = ‖ y , in other words, when the unit sphere of X does not contain non‐trivial segments. Our aim in this paper is the study of those normed spaces which admit an equivalent strictly convex norm. We present a characterization in linear topological terms of the normed spaces which are strictly convex renormable. We consider the class of all solid Banach lattices made up of bounded real functions on some set Γ. This class contains the Mercourakis space c 1 (Σ′ × Γ) and all duals of Banach spaces with unconditional uncountable bases. We characterize the elements of this class which admit a pointwise strictly convex renorming.