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Hyperplanes in the Space of Convergent Sequences and Preduals of ℓ1

Published online by Cambridge University Press:  20 November 2018

Emanuele Casini
Affiliation:
Dipartimento di Scienza e Alta Tecnologia, Università dell’Insubria, via Valleggio 11, 22100 Como, Italy e-mail: emanuele.casini@uninsubria.it
Enrico Miglierina
Affiliation:
Dipartimento di Discipline Matematiche, Finanza Matematica ed Econometria, Università Cattolica del Sacro Cuore, Via Necchi 9, 20123 Milano, Italy e-mail: enrico.miglierina@unicatt.it
Łukasz Piasecki
Affiliation:
Instytut Matematyki, Uniwersytet Marii Curie-Skłodowskiej, Pl. Marii Curie-Skłodowskiej 1, 20-031 Lublin, Poland e-mail: piasecki@hektor.umcs.lublin.pl
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Abstract

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The main aim of this paper is to investigate various structural properties of hyperplanes of $c$, the Banach space of the convergent sequences. In particular, we give an explicit formula for the projection constants and prove that an hyperplane of $c$ is isometric to the whole space if and only if it is 1-complemented. Moreover, we obtain the classification of those hyperplanes for which their duals are isometric to ${{\ell }_{1}}$ and give a complete description of the preduals of ${{\ell }_{1}}$ under the assumption that the standard basis of ${{\ell }_{1}}$ is weak$^{*}$-convergent.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2015

References

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