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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.
UDC:
517.98 Received: 11.10.2017
DOI:
10.4213/faa3534
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