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Continuous linear extension of functions
Author(s):
A.
Koyama;
I.
Stasyuk;
E.
D.
Tymchatyn;
A.
Zagorodnyuk
Journal:
Proc. Amer. Math. Soc.
138
(2010),
4149-4155.
MSC (2010):
Primary 54C20, 54C30;
Secondary 54E40
Posted:
May 26, 2010
MathSciNet review:
2679637
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Abstract:
Let  be a complete metric space. We prove that there is a continuous, linear, regular extension operator from the space  of all partial, continuous, real-valued, bounded functions with closed, bounded domains in  to the space  of all continuous, bounded, real-valued functions on  with the topology of uniform convergence on compact sets. This is a variant of a result of Kunzi and Shapiro for continuous functions with compact, variable domains.
References:
-
- 1.
- S. Ageev and E. D. Tymchatyn, On exact Milyutin maps, Topology Appl. 153 (2005), 227-238. MR 2175347 (2006h:54008)
- 2.
- G. Beer, Metric spaces on which continuous functions are uniformly continuous and Hausdorff distance, Proc. Amer. Math. Soc. 95 (1985), 653-658. MR 810180 (87e:54024)
- 3.
- T. Banakh, N. Brodskiy. I. Stasyuk and E. D. Tymchatyn, On continuous extension of uniformly continuous functions and metrics, Colloq. Math. 116 (2009), 191-202. MR 2520139
- 4.
- J. Dugundji, An extension of Tietze's theorem, Pacific J. Math. 1 (1951), 353-367. MR 0044116 (13:373c)
- 5.
- R. Engelking, General Topology, Heldermann, Berlin, 1989. MR 1039321 (91c:54001)
- 6.
- H. P. Kunzi and L. B. Shapiro, On simultaneous extension of continuous partial functions, Proc. Amer. Math. Soc. 125 (1997), 1853-1859. MR 1415348 (98g:54015)
- 7.
- K. Kuratowski, Sur l'espace des fonctions partielles, Ann. Mat. Pura Appl. 40 (1955), 61-67. MR 0074807 (17:650b)
- 8.
- E. Michael, Selected selection theorems, Amer. Math. Monthly 63 (1956), 233-238. MR 1529282
- 9.
- S. A. Naimpally, Graph topology for function spaces, Trans. Amer. Math. Soc. 123 (1966), 267-272. MR 0192466 (33:691)
- 10.
- L. Narici and E. Beckenstein, Topological vector spaces, Pure and Applied Mathematics 95, Marcel Dekker Inc., New York-Basel, 1985. MR 812056 (87c:46003)
- 11.
- A. Pelczynski, Linear extensions, linear averagings, and their applications to linear topological classification of spaces of continuous functions, Dissertationes Math. (Rozprawy Mat.) 58 (1968). MR 0227751 (37:3335)
- 12.
- E. N. Stepanova, Continuation of continuous functions and the metrizability of paracompact p-spaces, Mat. Zametki 53 (1993), 92-201. Translation in Math. Notes 53 (1993), 308-314. MR 94k:54031
- 13.
- E. D. Tymchatyn and M. Zarichnyi, On simultaneous linear extensions of partial (pseudo) metrics, Proc. Amer. Math. Soc. 132 (2004), 2799-2807. MR 2054807 (2005d:54032)
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Additional Information:
A.
Koyama
Affiliation:
Faculty of Science, Shizuoka University, 836 Ohya 422-8059, Shizuoka, Japan
Email:
sakoyam@ipc.shizuoka.ac.jp
I.
Stasyuk
Affiliation:
Department of Mechanics and Mathematics, Lviv National University, Universytetska St. 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, P1B 8L7, Canada
Email:
i_stasyuk@yahoo.com
E.
D.
Tymchatyn
Affiliation:
Department of Mathematics and Statistics, University of Saskatchewan, McLean Hall, 106 Wiggins Road, Saskatoon, SK S7N 5E6, Canada
Email:
tymchat@math.usask.ca
A.
Zagorodnyuk
Affiliation:
Institute for Applied Problems of Mechanics and Mathematics, Ukrainian Academy of Sciences, 3b Naukova St., Lviv 79060, Ukraine
Address at time of publication:
Prycarpathian National University, Ivano-Frankivsk, Ukraine
Email:
andriyzag@yahoo.com
DOI:
10.1090/S0002-9939-2010-10424-0
PII:
S 0002-9939(2010)10424-0
Keywords:
Extension of functions,
continuous linear operator,
metric space
Received by editor(s):
September 10, 2009
Received by editor(s) in revised form:
November 20, 2009 and February 3, 2010
Posted:
May 26, 2010
Additional Notes:
The second, third, and fourth authors were supported in part by NSERC grant No. OGP 0005616
Communicated by:
Nigel J. Kalton
Copyright of article:
Copyright
2010,
American Mathematical Society
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