Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-23T02:54:24.858Z Has data issue: false hasContentIssue false
  • English
  • Français

Necessary and sufficient conditions for precompact sets to be metrisable

Published online by Cambridge University Press:  17 April 2009

J.C. Ferrando ,
J. Kasakol  and
M. López Pellicer
J.C. Ferrando
Affiliation:
Centro de Investigación Operativa, Universidad Miguel Hernández, E-03202 Elche (Alicante), Spain, e-mail: [email protected]
J. Kasakol
Affiliation:
Faculty of Mathematics and Informatics, A. Mickiewicz University, 61-614 Poznań, Poland, e-mail: [email protected]
M. López Pellicer
Affiliation:
Departamento de Matemática Aplicada and IMPA, Universidad Politécnica, E-46022 Valencia, Spain, e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This self-contained paper characterises those locally convex spaces whose (weakly) precompact (respectively, compact) subsets are metrisable. Applications and examples are provided. Our approach also applies to get Cascales-Orihuela's, Valdivia's and Robertson's metrisation theorems for (pre)compact sets.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 74 , Issue 1 , August 2006 , pp. 7 - 13
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Arkhangel'skii, A.V., Topological function spaces, Mathematics and its Applications (Soviet Series) 78 (Kluwer Academic Publishers Group, Dordrecht, 1992).CrossRefGoogle Scholar
[2]Arkhangel'skii, A.V., General Topology III (Springer-Verlag, Berlin, Heidelberg, New York, 1995).Google Scholar
[3]Buchwalter, H. and Schmets, J., ‘Sur quelques propiétés de l'espace C s(T)’, J. Math. Pures Appl. 52 (1973), 337–352.Google Scholar
[4]Cascales, B., ‘On K-analytic locally convex spaces’, Arch. Math. (Basel) 49 (1987), 232–244.Google Scholar
[5]Cascales, B. and Orihuela, J., ‘Metrizability of precompact subsets in (LF)-spaces’, Proc. Edinburgh Math. Soc. 103 (1986), 293–299.Google Scholar
[6]Cascales, B. and Orihuela, J., ‘On compactness in locally convex spaces’, Math. Z. 195 (1987), 365–381.CrossRefGoogle Scholar
[7]Cascales, B., Kakol, J. and Saxon, S.A., ‘Weight of precompact sets and tightness’, J. Math. Anal. Appl. 269 (2002), 500–518.CrossRefGoogle Scholar
[8]Cascales, B., Kakol, J. and Saxon, S.A., ‘Metrizability vs. Freéchet-Urysohn property’, Proc. Amer. Math. Soc. 131 (2003), 3623–3631.CrossRefGoogle Scholar
[9]Ferrando, J. C., Kakol, J., López Pellicer, M. and Saxon, S.A., ‘Tightness and distinguished Fréchet spaces’, J. Math. Anal. Appl. (to appear).Google Scholar
[10]Floret, K., ‘Some aspects of the theory of locally convex inductive limits’, in Functional Analysis: Surveys and Recent Results, II, Proc. Conf. Functional Anal. Univ. Padeborn, Padeborn 1979 (North-Holland, Amsterdam, 1980), pp. 205–237.Google Scholar
[11]Floret, K., Weakly compact sets, Lecture Notes in Math. 801 (Springer-Verlag, Berlin, 1980).CrossRefGoogle Scholar
[12]Gullick, D. and Schmets, J., ‘Separability and semi-norm separability for spaces of bounded continuous functions’, Bull. Roy. Sci. Liége 41 (1972), 254–260.Google Scholar
[13]Kakol, J. and Saxon, S.A., ‘Montel (DF) -spaces, sequential (LM)-spaces and the strongest locally topology’, J. London Math. Soc. 66 (2002), 388–406.Google Scholar
[14]Künzi, H.P.A., Mršević, M., Reilly, I.L. and Vamanamurthy, M.K., ‘Pre-Lindelöf quasi-pseudo-metric and quasi-uniform spaces’, Mat. Vesnik. 46 (1994), 81–87.Google Scholar
[15]Pfister, H.H., ‘Bemerkungen zum Satz über die separabilität der Fréchet-Montel Raüme’, Arch. Math. (Basel) 27 (1976), 86–92.Google Scholar
[16]Robertson, N., ‘The metrisability of precompact sets’, Bull. Austral. Math. Soc. 43 (1991), 131–135.Google Scholar
[17]Talagrand, M., ‘Espaces de Banach faiblement K-analytiques’, Ann. Math. 110 (1979), 407–438.Google Scholar
[18]Valdivia, M., Topics in locally convex spaces, Notas de Matemática 85 (North Holland, Amsterdam, 1982).Google Scholar