Hyperplanes in the space of convergent sequences and preduals of ${\ell }_1$

Abstract

The main aim of the present 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 we 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 we give a complete description of the preduals of ${\ell }_1$ under the assumption that the standard basis of ${\ell }_1$ is weak${}^{\ast }$-convergent.