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On Cm-bounding sets

Part of: Calculus on manifolds; nonlinear operators Topological linear spaces and related structures

Published online by Cambridge University Press:  09 April 2009

Peter Biström ,
Sten Bjon  and
Mikael Lindström
Peter Biström
Affiliation:
Abo Akademi SF-20500 Abo Finland
Sten Bjon
Affiliation:
Abo Akademi SF-20500 Abo Finland
Mikael Lindström
Affiliation:
Abo Akademi SF-20500 Abo Finland
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Abstract

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Let E be a quasi-complete locally convex space and A a subset of E. It is shown that if every real-valued C∞-function in the weak topology of E is bounded on A, then A is relatively weakly compact. Furthermore, if all real-valued C∞-functions on E are bounded on A, then A is relatively compact in the associated semi-weak topology of E.

Keywords

Cm-bounding setrelatively compact setsemi-weak topologyinterchangeable double limit property.

MSC classification

Secondary: 46A99: None of the above, but in this section 58C25: Differentiable maps
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 54 , Issue 1 , February 1993 , pp. 20 - 28
Copyright
Copyright © Australian Mathematical Society 1993

References

[1]Aliprantis, C. and Burkinshaw, O., Positive Operators (Academic Press, San Diego, 1985).Google Scholar
[2]Biström, P., Bjon, S., and Lindström, M., ‘Function algebras on which homomorphisms are point evaluations on sequences’, Manuscripta Math. 73 (1991), 179–185.CrossRefGoogle Scholar
[3]Bourgain, J. and Diestel, J., ‘Limited operators and strict consingularity’, Math. Nachr. 119 (1984), 55–58.Google Scholar
[4]Dazord, J., ‘Factoring operators through c0’, Math. Ann. 220 (1976), 105–122.CrossRefGoogle Scholar
[5]Diestel, J., Sequences and Series in Banach spaces, Graduate Texts in Math. 97 (Springer, Berlin, 1984).Google Scholar
[6]Dineen, S., ‘Unbounded holomorphic functions on a Banach space’, J. London Math. Soc. 4 (1972), 461–465.CrossRefGoogle Scholar
[7]Emmanuele, G., ‘A dual characterization of Banach spaces not containing l1’, Bull. Acad. Polon. Sci. Sér. Sci. Math. 34 (1986), 155–159.Google Scholar
[8]Floret, K., Weakly compact sets, Lecture Notes in Math. 801 (Springer, Berlin, 1980).Google Scholar
[9]Howard, J., ‘Weak sequential denseness in Banach spaces’, Proc. Amer. Math. Soc. 99 (1987), 351–352 and 104, 334 (1988).Google Scholar
[10]Jarchow, H., Locally convex spaces (Teubner, Stuttgard, 1981).CrossRefGoogle Scholar
[11]Kriegl, A. and Nel, L., ‘Convenient vector spaces of smooth functions’, Math. Nachr. 147 (1990), 39–45.CrossRefGoogle Scholar
[12]Lindström, M. and Schlumprecht, Th., ‘On limitedness in locally convex spaces’, Arch. Math. (Basel) 53 (1989), 65–74.Google Scholar
[13]Lloyd, J., ‘Smooth partitions of unity on manifolds’, Trans. Amer. Math. Soc. 197 (1974), 249–259.Google Scholar
[14]Carreras, P. Pérez and Bonet, J., Barrelled locally convex spaces (North-Holland, Amsterdam, 1987).Google Scholar
[15]Valdivia, M., ‘Some new results on weak compactness’, J. Funct. Anal. 24 (1977), 1–10.CrossRefGoogle Scholar