Hyperplanes in the space of convergent sequences and preduals of $\ell_1$

TL;DR

This paper explores the structure of hyperplanes in the space of convergent sequences, providing explicit formulas for projection constants, characterizations of isometric hyperplanes, and classifications related to preduals of ll_1.

Contribution

It offers new explicit formulas, characterizations, and classifications of hyperplanes in the space of convergent sequences and their duals, enhancing understanding of their geometric and duality properties.

Findings

Hyperplanes of c are isometric to c iff they are 1-complemented.

Explicit formulas for projection constants of hyperplanes.

Classification of hyperplanes with duals isometric to ll_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$^{*}$-convergent.