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On simultaneous linear extensions of partial (pseudo)metrics

Authors: E. D. Tymchatyn and M. Zarichnyi
Journal: Proc. Amer. Math. Soc. 132 (2004), 2799-2807
MSC (2000): Primary 54E35, 54C20, 54E40
Published electronically: April 21, 2004
MathSciNet review: 2054807
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Abstract: We consider the question of simultaneous extension of (pseudo) metrics defined on nonempty closed subsets of a compact metrizable space. The main result is a counterpart of the result due to Künzi and Shapiro for the case of extension operators of partial continuous functions and includes, as a special case, Banakh's theorem on linear regular operators extending (pseudo)metrics.

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  • 1. T. Banakh, On linear regular operators extending (pseudo)metrics, preprint.
  • 2. T. Banakh, $\rm AE(0)$-spaces and regular operators extending (averaging) pseudometrics, Bull. Polish Acad. Sci. Math. 42 (1994), no. 3, 197-206.
  • 3. T. Banakh and C. Bessaga, On linear operators extending (pseudo)metrics, Bull. Polish Acad. Sci. Math. 48 (2000), no. 1, 35-49. MR 2001e:54023
  • 4. C. Bessaga, On linear operators and functors extending pseudometrics, Fund. Math. 142 (1993), no. 2, 101-122. MR 94h:54033
  • 5. C. Bessaga, Functional analytic aspects of geometry. Linear extending of metrics and related problems, Progress in Functional Analysis (Peñíscola, 1990), 247-257, North-Holland Math. Stud., 170, North-Holland, Amsterdam, 1992. MR 93a:54015
  • 6. R. H. Bing, Extending a metric, Duke Math. J. 14 (1947), 511-519. MR 9:521c
  • 7. J. Dugundji, An extension of Tietze's theorem, Pacific J. Math. 1 (1951). 353-367. MR 13:373c
  • 8. N. Dunford and J. T. Schwartz, Linear operators. Part I. General theory, with the assistance of William G. Bade and Robert G. Bartle. Reprint of the 1958 original. Wiley Classics Library. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1988. xiv+858 pp. MR 22:8302
  • 9. V. V. Filippov, Topological structure of solution spaces of ordinary differential equations (in Russian), Uspekhi Mat. Nauk 48 (1993), 103-154; English transl., Russian Math. Surveys 48 (1993), no. 1, 101-154. MR 94f:34008
  • 10. A. Fryszkowski, Continuous selections for a class of non-convex multi-valued maps, Studia Math. 76 (1983), 163-174. MR 85j:54022
  • 11. M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, Based on the 1981 French original [MR 85e:53051]. With appendices by M. Katz, P. Pansu and S. Semmes. Translated from the French by Sean Michael Bates. Progress in Mathematics 152, Birkhäuser Boston, Inc., Boston, MA, 1999. xx+585 pp. MR 2000d:53065
  • 12. F. Hausdorff, Erweiterung einer Homöomorphie, Fund. Math. 16 (1930), 353-360.
  • 13. F. Hiai and H. Umegaki, Integrals, conditional expectations and martingales of multivalued functions, J. Multivariate Anal. 7 (1977), 149-182. MR 58:22463
  • 14. K. Kuratowski, Sur l'espace des fonctions partielles, Ann. Mat. Pura Appl. 40 (1955), 61-67. MR 17:650b
  • 15. K. Kuratowski, Sur une méthode de métrisation complète de certains espaces d'ensembles compacts, Fund. Math. 43 (1956), 114-138. MR 18:58a
  • 16. R. T. Rockafellar, Integrals which are convex functionals, Pacific J. Math. 24 (1968), 525-539. MR 38:4984
  • 17. H.-P. Künzi and L. B. Shapiro, On simultaneous extension of continuous partial functions, Proc. Amer. Math. Soc. 125 (1997), 1853-1859. MR 98g:54015
  • 18. Nguyen Van Khue and Nguyen To Nhu, Two extensors of metrics, Bull. Acad. Polon. Sci. Sér. Sci. Math. 29 (1981), no. 5-6, 285-291. MR 83g:54040
  • 19. J. Luukkainen, Extension of locally uniformly equivalent metrics, Colloq. Math. 46 (1982), no. 2, 205-207. MR 84a:54054
  • 20. O. Pikhurko, Extending metrics in compact pairs, Mat. Stud. 3 (1994), 103-106, 122.
  • 21. E. N. Stepanova, Continuation of continuous functions and the metrizability of paracompact $p$-spaces (Russian) Mat. Zametki 53 (1993), no. 3, 92-101; translation in Math. Notes 53 (1993), no. 3-4, 308-314. MR 94k:54031
  • 22. H. Torunczyk, A short proof of Hausdorff's theorem on extending metrics, Fund. Math. 77 (1972), no. 2, 191-193. MR 47:9559
  • 23. M. Zarichnyi, Regular linear operators extending metrics: a short proof, Bull. Polish Acad. Sci. Math. 44 (1996), no. 3, 267-269. MR 97m:54050

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Additional Information

E. D. Tymchatyn
Affiliation: Department of Mathematics and Statistics, University of Saskatchewan, Room 142 McLean Hall, 106 Wiggins Road, Saskatoon, Saskatchewan, Canada S7N 5E6

M. Zarichnyi
Affiliation: Department of Mechanics and Mathematics, Lviv National University, Universytetska 1, 79000 Lviv, Ukraine

Received by editor(s): November 18, 2002
Received by editor(s) in revised form: May 29, 2003
Published electronically: April 21, 2004
Additional Notes: The paper was finished during the second named author’s visit to the University of Saskatchewan. This research was supported in part by NSERC research grant OGP 005616
Communicated by: Alan Dow
Article copyright: © Copyright 2004 American Mathematical Society

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