, Volume 17, Issue 2, pp 257-263,
Open Access This content is freely available online to anyone, anywhere at any time.
Date: 26 Feb 2012

The positive aspects of smoothness in Banach lattices

Abstract

Let X be a Banach lattice, and let \({x\in X{\setminus}\{0\}}\) . We study the structure of the set Grad(x), of all supporting functionals of x. If X is a Dedekind σ-complete Banach lattice, there is an isometry from Grad(x) onto Grad(|x|); hence the elements x and |x| are smooth simultaneously. And if, additionally, X* is strictly monotone then Grad(|x|) consists of positive functionals. As a by-product of our results we obtain that an arbitrary Banach lattice X is strictly monotone whenever its dual X* is smooth.