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Necessary and sufficient conditions for precompact sets to be metrisable

  • J.C. Ferrando (a1), J. Kasakol (a2) and M. López Pellicer (a3)
Abstract

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.

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References
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[1]Arkhangel'skii A.V., Topological function spaces, Mathematics and its Applications (Soviet Series) 78 (Kluwer Academic Publishers Group, Dordrecht, 1992).
[2]Arkhangel'skii A.V., General Topology III (Springer-Verlag, Berlin, Heidelberg, New York, 1995).
[3]Buchwalter H. and Schmets J., ‘Sur quelques propiétés de l'espace C s(T)’, J. Math. Pures Appl. 52 (1973), 337352.
[4]Cascales B., ‘On K-analytic locally convex spaces’, Arch. Math. (Basel) 49 (1987), 232244.
[5]Cascales B. and Orihuela J., ‘Metrizability of precompact subsets in (LF)-spaces’, Proc. Edinburgh Math. Soc. 103 (1986), 293299.
[6]Cascales B. and Orihuela J., ‘On compactness in locally convex spaces’, Math. Z. 195 (1987), 365381.
[7]Cascales B., Kakol J. and Saxon S.A., ‘Weight of precompact sets and tightness’, J. Math. Anal. Appl. 269 (2002), 500518.
[8]Cascales B., Kakol J. and Saxon S.A., ‘Metrizability vs. Freéchet-Urysohn property’, Proc. Amer. Math. Soc. 131 (2003), 36233631.
[9]Ferrando J. C., Kakol J., López Pellicer M. and Saxon S.A., ‘Tightness and distinguished Fréchet spaces’, J. Math. Anal. Appl. (to appear).
[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. 205237.
[11]Floret K., Weakly compact sets, Lecture Notes in Math. 801 (Springer-Verlag, Berlin, 1980).
[12]Gullick D. and Schmets J., ‘Separability and semi-norm separability for spaces of bounded continuous functions’, Bull. Roy. Sci. Liége 41 (1972), 254260.
[13]Kakol J. and Saxon S.A., ‘Montel (DF) -spaces, sequential (LM)-spaces and the strongest locally topology’, J. London Math. Soc. 66 (2002), 388406.
[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), 8187.
[15]Pfister H.H., ‘Bemerkungen zum Satz über die separabilität der Fréchet-Montel Raüme’, Arch. Math. (Basel) 27 (1976), 8692.
[16]Robertson N., ‘The metrisability of precompact sets’, Bull. Austral. Math. Soc. 43 (1991), 131135.
[17]Talagrand M., ‘Espaces de Banach faiblement K-analytiques’, Ann. Math. 110 (1979), 407438.
[18]Valdivia M., Topics in locally convex spaces, Notas de Matemática 85 (North Holland, Amsterdam, 1982).
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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