Abstract
We study the geometry of the unit ball of ℓ∞(Λ) and of the dual space, proving, among other things, that Λ is countable if and only if 1 is an exposed point of \({B_{{\ell _\infty }\left( \Lambda \right)}}\). On the other hand, we prove that Λ is finite if and only if the δλ are the only functionals taking the value 1 at a canonical element and vanishing at all other canonical elements. We also show that the restrictions of evaluation functionals to a 2-dimensional subspace are not necessarily extreme points of the dual of that subspace. Finally, we prove that if Λ is uncountable, then the face of \({B_{{\ell _\infty }\left( \Lambda \right)*}}\) consisting of norm 1 functionals attaining their norm at the constant function 1 has empty interior relative to \({S_{{\ell _\infty }\left( \Lambda \right)*}}\).
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Translated from Funktsional′nyi Analiz i Ego Prilozheniya, Vol. 52, No. 4, pp. 62–71, 2018
The author was supported by Research Grant MTM2014-58984-P (this project has been funded by the Spanish Ministry of Economy and Competitivity and by the European Fund for Regional Development FEDER).
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García-Pacheco, F.J. Cardinality of Λ Determines the Geometry of \({B_{{\ell _\infty }\left( \Lambda \right)}}\) and \({B_{{\ell _\infty }\left( \Lambda \right)*}}\). Funct Anal Its Appl 52, 290–296 (2018). https://doi.org/10.1007/s10688-018-0238-z
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DOI: https://doi.org/10.1007/s10688-018-0238-z