# The positive aspects of smoothness in Banach lattices

The positive aspects of smoothness in Banach lattices 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# The positive aspects of smoothness in Banach lattices

, Volume 17 (2) – Feb 26, 2012
7 pages

/lp/springer_journal/the-positive-aspects-of-smoothness-in-banach-lattices-0A0UeTTGOU
Publisher
SP Birkhäuser Verlag Basel
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-012-0163-y
Publisher site
See Article on Publisher Site

### 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.

### Journal

PositivitySpringer Journals

Published: Feb 26, 2012

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