Positivity

, Volume 17, Issue 2, pp 257–263

The positive aspects of smoothness in Banach lattices

Open Access
Article

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.

Keywords

Smooth point Strictly monotone Banach lattice Orlicz space Marcinkiewicz space M-ideal 

Mathematics Subject Classification (2010)

46B20 46B42 

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Copyright information

© The Author(s) 2012

Authors and Affiliations

  1. 1.Instytut MatematykiUniwersytet Kazimierza WielkiegoBydgoszczPoland

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