Continuous linear extension of functions

Authors:
A. Koyama, I. Stasyuk, E. D. Tymchatyn and A. Zagorodnyuk

Journal:
Proc. Amer. Math. Soc. **138** (2010), 4149-4155

MSC (2010):
Primary 54C20, 54C30; Secondary 54E40

Published electronically:
May 26, 2010

MathSciNet review:
2679637

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**1.**S. Ageev and E. D. Tymchatyn,*On exact atomless Milutin maps*, Topology Appl.**153**(2005), no. 2-3, 227–238. MR**2175347**, 10.1016/j.topol.2003.07.021**2.**Gerald Beer,*Metric spaces on which continuous functions are uniformly continuous and Hausdorff distance*, Proc. Amer. Math. Soc.**95**(1985), no. 4, 653–658. MR**810180**, 10.1090/S0002-9939-1985-0810180-3**3.**T. Banakh, N. Brodskiy, I. Stasyuk, and E. D. Tymchatyn,*On continuous extension of uniformly continuous functions and metrics*, Colloq. Math.**116**(2009), no. 2, 191–202. MR**2520139**, 10.4064/cm116-2-4**4.**J. Dugundji,*An extension of Tietze’s theorem*, Pacific J. Math.**1**(1951), 353–367. MR**0044116****5.**Ryszard Engelking,*General topology*, 2nd ed., Sigma Series in Pure Mathematics, vol. 6, Heldermann Verlag, Berlin, 1989. Translated from the Polish by the author. MR**1039321****6.**Hans-Peter A. Künzi and Leonid B. Shapiro,*On simultaneous extension of continuous partial functions*, Proc. Amer. Math. Soc.**125**(1997), no. 6, 1853–1859. MR**1415348**, 10.1090/S0002-9939-97-04011-2**7.**K. Kuratowski,*Sur l’espace des fonctions partielles*, Ann. Mat. Pura Appl. (4)**40**(1955), 61–67 (French). MR**0074807****8.**E. Michael,*Selected Selection Theorems*, Amer. Math. Monthly**63**(1956), no. 4, 233–238. MR**1529282**, 10.2307/2310346**9.**Somashekhar Amrith Naimpally,*Graph topology for function spaces*, Trans. Amer. Math. Soc.**123**(1966), 267–272. MR**0192466**, 10.1090/S0002-9947-1966-0192466-4**10.**Lawrence Narici and Edward Beckenstein,*Topological vector spaces*, Monographs and Textbooks in Pure and Applied Mathematics, vol. 95, Marcel Dekker, Inc., New York, 1985. MR**812056****11.**A. Pełczyński,*Linear extensions, linear averagings, and their applications to linear topological classification of spaces of continuous functions*, Dissertationes Math. Rozprawy Mat.**58**(1968), 92. MR**0227751****12.**E. N. Stepanova,*Continuation of continuous functions and the metrizability of paracompact 𝑝-spaces*, Mat. Zametki**53**(1993), no. 3, 92–101 (Russian); English transl., Math. Notes**53**(1993), no. 3-4, 308–314. MR**1220188**, 10.1007/BF01207718**13.**E. D. Tymchatyn and M. Zarichnyi,*On simultaneous linear extensions of partial (pseudo)metrics*, Proc. Amer. Math. Soc.**132**(2004), no. 9, 2799–2807. MR**2054807**, 10.1090/S0002-9939-04-07413-1

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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:
http://dx.doi.org/10.1090/S0002-9939-2010-10424-0

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

Published electronically:
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

Article copyright:
© Copyright 2010
American Mathematical Society