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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 continuous linear operators extending metrics


Authors: I. Stasyuk and E. D. Tymchatyn
Journal: Proc. Amer. Math. Soc. 141 (2013), 2913-2921
MSC (2010): Primary 54C20, 54C30; Secondary 54E40
Published electronically: April 24, 2013
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Abstract: Let $ (X,d)$ be a complete metric space. We prove that there is a continuous, linear extension operator from the space of all partial, continuous, bounded metrics with closed, bounded domains in $ X$ endowed with the Hausdorff metric topology to the space of all continuous, bounded, metrics on $ X$ with the topology of uniform convergence on compact sets. This is a variant of the result of Tymchatyn and Zarichnyi for continuous metrics defined on closed, variable domains in a compact metric space. We get a similar result for the case of continuous real-valued functions.


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Additional Information

I. Stasyuk
Affiliation: Department of Mechanics and Mathematics, Lviv National University, Universytetska Str. 1, Lviv, 79000, Ukraine
Address at time of publication: Department of Computer Science and Mathematics, Nipissing University, 100 College Drive, Box 5002, North Bay, ON, 51B 8L7, Canada
Email: i$\textunderscore$stasyuk@yahoo.com

E. D. Tymchatyn
Affiliation: Department of Mathematics and Statistics, McLean Hall, University of Saskatchewan, 106 Wiggins Road, Saskatoon, SK S7N 5E6, Canada
Email: tymchat@math.usask.ca

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11547-9
PII: S 0002-9939(2013)11547-9
Keywords: Extension of metrics, continuous linear operator, metric space
Received by editor(s): October 14, 2010
Received by editor(s) in revised form: November 8, 2011
Published electronically: April 24, 2013
Additional Notes: The authors were supported in part by NSERC grant No. OGP 0005616
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.