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Positivity

, Volume 20, Issue 1, pp 171–185 | Cite as

Generalized Dini theorems for nets of functions on arbitrary sets

  • Vlad TimofteEmail author
  • Aida Timofte
Article

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.

Keywords

Dini theorem Uniform convergence Pointwise convergence  Net of functions Topological ordered vector space Stone-Čech compactification 

Mathematics Subject Classification

46A40 40A30 

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Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Institute of Mathematics “Simion Stoilow” of the Romanian AcademyBucharestRomania

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