A note on the Banach lattice $c_0( \ell_2^n)$, its dual and its bidual

TL;DR

This paper investigates the geometric and topological properties of the Banach lattice c_0(ell_2^n), its dual, and bidual, revealing specific properties like the strong Gelfand-Philips property and the nature of the unit ball in l_infinity(ell_2^n).

Contribution

It provides new insights into the structure of c_0(ell_2^n), including its properties related to the Gelfand-Philips and Grothendieck properties, and characterizes the unit ball in l_infinity(ell_2^n).

Findings

c_0(ell_2^n) has the strong Gelfand-Philips property

c_0(ell_2^n) does not have the positive Grothendieck property

The unit ball of l_infinity(ell_2^n) is an almost limited set

Abstract

The main purpose of this paper is to study some geometric and topological properties on $ c_0$ sum of the finite dimensional Banach lattice $\ell_2^n$, its dual and its bidual. Among other results, we show that the Banach lattices $c_0(\ell_2^n)$ has the strong Gelfand-Philips property, but does not have the positive Grothendieck property. We also prove that the closed unit ball of $l_{\infty}(\ell_2^n)$ is an almost limited set.