On simultaneous extension of continuous partial functions
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- by Hans-Peter A. Künzi and Leonid B. Shapiro PDF
- Proc. Amer. Math. Soc. 125 (1997), 1853-1859 Request permission
Abstract:
For a metric space $X$ let $\mathcal {C}_{vc}(X)$ (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 $\Phi :\mathcal {C}_{vc}(X)\rightarrow \mathcal {C}_b(X)$ 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 $X,$ $\Phi \vert \mathcal {C}_b(K):\mathcal {C}_b(K) \rightarrow \mathcal {C}_b(X)$ is a linear positive operator such that $\Phi (1_K)=1_X$.References
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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
- 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
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 1853-1859
- MSC (1991): Primary 54B20, 54C20, 54C35, 54C65, 54E15
- DOI: https://doi.org/10.1090/S0002-9939-97-04011-2
- MathSciNet review: 1415348