Generalized Dini theorems for nets of functions on arbitrary sets

Generalized Dini theorems for nets of functions on arbitrary sets We characterize the uniform convergence of pointwise monotonic nets of bounded real functions defined on arbitrary sets, without any particular structure. The resulting condition trivially holds for the classical Dini theorem. Our vector-valued Dini-type theorem characterizes the uniform convergence of pointwise monotonic nets of functions with relatively compact range in Hausdorff topological ordered vector spaces. As a consequence, for such nets of continuous functions on a compact space, we get the equivalence between the pointwise and the uniform convergence. When the codomain is locally convex, we also get the equivalence between the uniform convergence and the weak-pointwise convergence; this also merges the Dini-Weston theorem on the convergence of monotonic nets from Hausdorff locally convex ordered spaces. Most of our results are free of any structural requirements on the common domain and put compactness in the right place: the range of the functions. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Generalized Dini theorems for nets of functions on arbitrary sets

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Publisher
Springer International Publishing
Copyright
Copyright © 2015 by Springer 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/s11117-015-0348-2
Publisher site
See Article on Publisher Site

Abstract

We characterize the uniform convergence of pointwise monotonic nets of bounded real functions defined on arbitrary sets, without any particular structure. The resulting condition trivially holds for the classical Dini theorem. Our vector-valued Dini-type theorem characterizes the uniform convergence of pointwise monotonic nets of functions with relatively compact range in Hausdorff topological ordered vector spaces. As a consequence, for such nets of continuous functions on a compact space, we get the equivalence between the pointwise and the uniform convergence. When the codomain is locally convex, we also get the equivalence between the uniform convergence and the weak-pointwise convergence; this also merges the Dini-Weston theorem on the convergence of monotonic nets from Hausdorff locally convex ordered spaces. Most of our results are free of any structural requirements on the common domain and put compactness in the right place: the range of the functions.

Journal

PositivitySpringer Journals

Published: Jul 8, 2015

References

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