On $\ell_\infty$-Grothendieck subspaces

TL;DR

This paper investigates the properties of certain subspaces of , called -Grothendieck subspaces, providing examples of when they do or do not satisfy specific convergence conditions.

Contribution

It introduces the concept of -Grothendieck subspaces and offers examples illustrating their existence and properties.

Findings

Examples of -Grothendieck subspaces that do and do not satisfy the property.

Characterization of these subspaces in relation to convergence of sequences.

Insights into the structure of subspaces of  with respect to dual space convergence.

Abstract

A closed subspace $S$ of $\ell_\infty$ is said to be a \emph{$\ell_\infty$-Grothendieck subspace} if $c_0\subset S$ (hence $\ell_\infty\subset S^{**}$) and every $\sigma(S^*,S)$-convergent sequence in $S^*$ is $\sigma(S^*,\ell_\infty)$-convergent. Here we give examples of closed subspaces of $\ell_\infty$ containing $c_0$ which are or fail to be $\ell_\infty$-Grothendieck.