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

Abstract

The main purpose of this paper is to study some geometric properties on c 0 sum of the finite dimensional Banach lattice, ℓ n 2 and its dual. Among other results, we show that the Banach lattices c 0 (ℓ n 2 ) has the strong Gelfand-Philips property, but does not have the positive Grothendieck property. We also prove that the closed unit ball of l ∞ (ℓ n 2 ) is an almost limited set.

Grothendieck-type subsets of Banach lattices

“…✷ In [15], the authors proved that every L-space has the weak Grothendieck property. Also in [14], it is proved that the Banach lattice ( n∈N ℓ n…”

“…2 ) has the weak Grothendieck property. By using the same argument as in Proposition 2.8 of [14], we have the following: Proposition 2.6 Assume that E ′ , the dual Banach lattice, has an order unit, i.e. there exists e ′ > 0 such that B E ′ = [−e ′ , e ′ ].…”