On quasi-metric and metric spaces
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- by Maciej Paluszyński and Krzysztof Stempak PDF
- Proc. Amer. Math. Soc. 137 (2009), 4307-4312 Request permission
Abstract:
Given a space $X$ with a quasi-metric $\rho$ it is known that the so-called $p$-chain approach can be used to produce a metric in $X$ equivalent to $\rho ^p$ for some $0<p\le 1$, hence also a quasi-metric $\tilde {\rho }$ equivalent to $\rho$ with better properties. We refine this result and obtain an exponent $p$ which is, in general, optimal.References
- Hugo Aimar, Bibiana Iaffei, and Liliana Nitti, On the Macías-Segovia metrization of quasi-metric spaces, Rev. Un. Mat. Argentina 41 (1998), no. 2, 67–75. MR 1700292
- Tosio Aoki, Locally bounded linear topological spaces, Proc. Imp. Acad. Tokyo 18 (1942), 588–594. MR 14182
- Michael Christ, Lectures on singular integral operators, CBMS Regional Conference Series in Mathematics, vol. 77, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1990. MR 1104656
- 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
- A. H. Frink, Distance functions and the metrization problem, Bull. Amer. Math. Soc. 43 (1937), no. 2, 133–142. MR 1563501, DOI 10.1090/S0002-9904-1937-06509-8
- Juha Heinonen, Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York, 2001. MR 1800917, DOI 10.1007/978-1-4613-0131-8
- N. J. Kalton, N. T. Peck, and James W. Roberts, An $F$-space sampler, London Mathematical Society Lecture Note Series, vol. 89, Cambridge University Press, Cambridge, 1984. MR 808777, DOI 10.1017/CBO9780511662447
- Roberto A. Macías and Carlos Segovia, Lipschitz functions on spaces of homogeneous type, Adv. in Math. 33 (1979), no. 3, 257–270. MR 546295, DOI 10.1016/0001-8708(79)90012-4
- J. Peetre and G. Sparr, Interpolation of normed abelian groups, Ann. Mat. Pura Appl. (4) 92 (1972), 217–262. MR 322529, DOI 10.1007/BF02417949
- S. Rolewicz, On a certain class of linear metric spaces, Bull. Acad. Polon. Sci. Cl. III. 5 (1957), 471–473, XL (English, with Russian summary). MR 0088682
Additional Information
- Maciej Paluszyński
- Affiliation: Instytut Matematyczny, Uniwersytet Wrocławski, Pl. Grunwaldzki 2/4, 50–384 Wrocław, Poland
- Email: mpal@math.uni.wroc.pl
- Krzysztof Stempak
- Affiliation: Instytut Matematyki i Informatyki, Politechnika Wrocławska, Wyb. Wyspiańskiego 27, 50–370 Wrocław, Poland – and – Katedra Matematyki i Zastosowań Informatyki, Politechnika Opolska, ul. Mikołajczyka 5, 45-271 Opole, Poland
- Email: Krzysztof.Stempak@pwr.wroc.pl
- Received by editor(s): January 18, 2009
- Received by editor(s) in revised form: May 12, 2009
- Published electronically: August 7, 2009
- Additional Notes: The authors’ research was supported in part by grants KBN #1P03A03029 and MNiSW #N201 054 32/4285, respectively.
- Communicated by: Nigel J. Kalton
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 137 (2009), 4307-4312
- MSC (2000): Primary 54E35; Secondary 54E15
- DOI: https://doi.org/10.1090/S0002-9939-09-10058-8
- MathSciNet review: 2538591