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Funktsional. Anal. i Prilozhen., 2018, Volume 52, Issue 4, Pages 62–71 (Mi faa3534)  

Cardinality of $\Lambda$ Determines the Geometry of $\mathsf{B}_{\ell_\infty(\Lambda)}$ and $\mathsf{B}_{\ell_\infty(\Lambda)^*}$

F. J. Garcia-Pacheco

Universidad de Cadiz

Abstract: We study the geometry of the unit ball of $\ell_\infty(\Lambda)$ and of the dual space, proving, among other things, that $\Lambda$ is countable if and only if $1$ is an exposed point of $\mathsf{B}_{\ell_\infty(\Lambda)}$. On the other hand, we prove that $\Lambda$ is finite if and only if the $\delta_\lambda$ 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 $\Lambda$ is uncountable, then the face of $\mathsf{B}_{\ell_\infty(\Lambda)^*}$ consisting of norm $1$ functionals attaining their norm at the constant function $1$ has empty interior relative to $\mathsf{S}_{\ell_\infty(\Lambda)^*}$.

Keywords: bounded functions, extremal structure.

Funding Agency Grant Number
Ministerio de Economía y Competitividad MTM2014-58984-P
Federación Española de Enfermedades Raras
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).


DOI: https://doi.org/10.4213/faa3534

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English version:
Functional Analysis and Its Applications, 2018, 52:4, 290–296

Bibliographic databases:

UDC: 517.98
Received: 11.10.2017

Citation: F. J. Garcia-Pacheco, “Cardinality of $\Lambda$ Determines the Geometry of $\mathsf{B}_{\ell_\infty(\Lambda)}$ and $\mathsf{B}_{\ell_\infty(\Lambda)^*}$”, Funktsional. Anal. i Prilozhen., 52:4 (2018), 62–71; Funct. Anal. Appl., 52:4 (2018), 290–296

Citation in format AMSBIB
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\jour Funktsional. Anal. i Prilozhen.
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\vol 52
\issue 4
\pages 62--71
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