# Integral Representations for Continuous Linear Functionals in Operator-Initiated Topologies

Integral Representations for Continuous Linear Functionals in Operator-Initiated Topologies On a given cone (resp. vector space) $$\mathcal{Q}$$ we consider an initial topology and order induced by a family of linear operators into a second cone $$\mathcal{P}$$ which carries a locally convex topology. We prove that monotone linear functionals on $$\mathcal{Q}$$ which are continuous with respect to this initial topology may be represented as certain integrals of continuous linear functionals on $$\mathcal{P}$$ . Based on the Riesz representation theorem from measure theory, we derive an integral version of the Jordan decomposition for linear functionals on ordered vector spaces. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Integral Representations for Continuous Linear Functionals in Operator-Initiated Topologies

, Volume 6 (2) – Oct 14, 2004
13 pages

/lp/springer_journal/integral-representations-for-continuous-linear-functionals-in-operator-PBQcWCCs4U
Publisher
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.
10.1023/A:1015236803527
Publisher site
See Article on Publisher Site

### Abstract

On a given cone (resp. vector space) $$\mathcal{Q}$$ we consider an initial topology and order induced by a family of linear operators into a second cone $$\mathcal{P}$$ which carries a locally convex topology. We prove that monotone linear functionals on $$\mathcal{Q}$$ which are continuous with respect to this initial topology may be represented as certain integrals of continuous linear functionals on $$\mathcal{P}$$ . Based on the Riesz representation theorem from measure theory, we derive an integral version of the Jordan decomposition for linear functionals on ordered vector spaces.

### Journal

PositivitySpringer Journals

Published: Oct 14, 2004

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