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This article is cited in 1 scientific paper (total in 1 paper)
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.
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
Linking options:
https://www.mathnet.ru/eng/faa3534https://doi.org/10.4213/faa3534 https://www.mathnet.ru/eng/faa/v52/i4/p62
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Abstract page: | 228 | Full-text PDF : | 25 | References: | 19 | First page: | 10 |
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