Research and Innovation Articles
Published 2018-07-22
Keywords
- C(K) Banach spaces,
- Banach-Stone theorem
How to Cite
Rincón-Villamizar, M. A. (2018). A proof of Holsztyński theorem. Revista Integración, Temas De matemáticas, 36(1), 59–65. https://doi.org/10.18273/revint.v36n1-2018005
Abstract
For a compact Hausdorff space, we denote by C(K) the Banach space of continuous functions defined in K with values in R or C. A well known result in Banach spaces of continuous functions is the Holsztyński theorem which establishes that if C(K) is isometric to a subspace of C(S), then K is a continuous image of S. The aim of this paper is to give an alternative proof of this result for extremely regular subspaces of C(K).
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References
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[2] Araujo J., Font J.J. and Hernández S., “A note on Holsztynski’s theorem”, Papers on general topology and applications, in Ann. New York Acad. Sci. 788, New York (1996), 9–12.
[3] Arens R.F. and Kelley J.L., “Characterization of the space of continuous functions over a compact Hausdorff space”, Trans. Amer. Math. Soc. 62 (1947), 499–508.
[4] Banach S., Théorie des opérations linéaires, Monografie Matematyczne, Warsaw, 1933.
[5] Behrends E., M-Sctructure and Banach-Stone theorem, Lecture Notes in Math. 736, Springer-Verlag, 1979.
[6] Cengiz B., “Continuous maps induced by isomorphisms of extremely regular function spaces”, J. Pure Appl. Sci. 18 (1985), No. 3, 377–384.
[7] Folland G.B., Real analysis, Modern techniques and their applications, Pure and Applied Mathematics (New York), A Wiley-Interscience Publication, John Wiley & Sons Inc., New York, 1999.
[8] Holsztyński W., “Continuous mappings induced by isometries of spaces of continuous functions”, Studia Math. 26 (1966), 133–136.
[9] Plebanek G., “On isomorphisms of Banach spaces of continuous functions”, Israel J. Math. 209 (2015), No. 1, 1–13.
[10] Semadeni Z., Banach spaces of Continuous Functions, Vol. 1, Monografie Mat. 55, PWN-Polish Sci. Publ. Warszawa, 1971.
[11] Stone M.H., “Applications of the theory of Boolean rings to general topology”, Trans. Amer. Math. Soc. 41 (1937), 375–481.