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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On simultaneous extension
of continuous partial functions


Authors: Hans-Peter A. Künzi and Leonid B. Shapiro
Journal: Proc. Amer. Math. Soc. 125 (1997), 1853-1859
MSC (1991): Primary 54B20, 54C20, 54C35, 54C65, 54E15
MathSciNet review: 1415348
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Abstract: For a metric space $X$ let ${\cal C}_{vc}(X)$ (that is, the set of all graphs of real-valued continuous functions with a compact domain in $X$) be equipped with the Hausdorff metric induced by the hyperspace of nonempty closed subsets of $X\times {\mathbf {R}}.$ It is shown that there exists a continuous mapping $\Phi :{\cal C}_{vc}(X)\rightarrow {\cal C}_b(X)$ satisfying the following conditions: (i) $\Phi (f)\vert \operatorname {dom}f= f$ for all partial functions $f.$ (ii) For every nonempty compact subset $K$ of $X,$ $\Phi \vert{\cal C}_b(K):{\cal C}_b(K) \rightarrow {\cal C}_b(X)$ is a linear positive operator such that $\Phi (1_K)=1_X$.


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Additional Information

Hans-Peter A. Künzi
Affiliation: Department of Mathematics, University of Berne, Sidlerstrasse 5, CH-3012 Berne, Switzerland
Email: kunzi@math-stat.unibe.ch

Leonid B. Shapiro
Affiliation: Department of Mathematics, Academy of Labor and Social Relations, Lobachevskogo 90, 117454 Moscow, Russia
Email: lshapiro@glas.apc.org

DOI: http://dx.doi.org/10.1090/S0002-9939-97-04011-2
PII: S 0002-9939(97)04011-2
Keywords: Extension of function, partial function, compact domain, Hausdorff metric, Lipschitzian function, probability measure
Received by editor(s): December 16, 1995
Additional Notes: The first author was partially working on this paper during his stay at the University of Łódź in 1995. He would like to thank his Polish colleagues for their hospitality.
During his visit to the University of Berne the second author was supported by the first author’s grant 7GUPJ041377 from the Swiss National Science Foundation and by the International Science Foundation under grants NFU 000 and NFU 300.
Communicated by: Franklin D. Tall
Article copyright: © Copyright 1997 American Mathematical Society