Hostname: page-component-5c6d5d7d68-xq9c7 Total loading time: 0 Render date: 2024-08-19T14:14:55.319Z Has data issue: false hasContentIssue false
  • English
  • Français

Weak compactness in normed linear spaces

Published online by Cambridge University Press:  09 April 2009

D. G. Tacon
D. G. Tacon
Affiliation:
Australian National UniversityCanberra, ACT
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.

The notion of weak compactness plays an important role in normed linear space theory. However in many instances norm completeness of the space is an important assumption for obtaining relevant theorems. This note generalises the concept of weak compactness for subsets of normed linear spaces and obtains generalisations of known theorems including that of Eberlein [2]. We use the methods and techniques of “non-standard analysis” in the proofs; several useful non-standard results for normed linear spaces are given.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 14 , Issue 1 , July 1972 , pp. 9 - 16
Copyright
Copyright © Australian Mathematical Society 1972

References

[1]Dunford, N. and Schwartz, J. T., Linear Operators, Part 1 (Interscience, New York, 1958).Google Scholar
[2]Eberlein, W. F., ‘Weak compactness in Banach spaces’, I, Proc. Nat. Acad. Sci. USA 33 (1947), 51–53.CrossRefGoogle Scholar
[3]Edwards, R. E., Functional Analysis: Theory and Applications (Holt, Rinehart and Winston Inc., New York, 1965).Google Scholar
[4]Fan, Ky and Glicksberg, Irving, ‘Fully convex normed linear spaces’, Proc. Nat. Acad. Sci. USA 41 (1955), 947–953.CrossRefGoogle ScholarPubMed
[5]Gustafson, Karl, ‘Compact-like operators and the Eberlein theorem’, Amer. Math. Monthly 75 (1968), 958–964.CrossRefGoogle Scholar
[6]Pelczynski, A., ‘A proof of the Eberlein-Šmulian theorem by an application of basic sequences’, Bull. Polon. Sci. Ser. Math. Astr. et Phys. 12 (1964), 543–548.Google Scholar
[7]Robinson, A., Non-standard Analysis (Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1966).Google Scholar
[8]Robinson, Abraham, ‘Non-standard arithmetic’, Bull. Amer. Math. Soc. 73 (1967), 818–843.CrossRefGoogle Scholar
[9]Staples, John, ‘A non-standard representation of Boolean algebras’, Bull. London Math. Soc. 1 (1969), 315–320.CrossRefGoogle Scholar
[10]Whitley, Robert, ‘An elementary proof of the Eberlein-Šmulian theorem’, Math. Annalen 172 (1967), 116–118.CrossRefGoogle Scholar
[11]Wilansky, A., Functional Analysis (Blaisdell, 1964).Google Scholar