Hostname: page-component-8448b6f56d-cfpbc Total loading time: 0 Render date: 2024-04-24T05:49:27.275Z Has data issue: false hasContentIssue false
  • English
  • Français

Extreme Points and Linear Isometries of the Banach Space of Lipschitz Functions

Published online by Cambridge University Press:  20 November 2018

Ashoke K. Roy
Ashoke K. Roy*
Affiliation:
Boston University, Boston, Massachusetts ; Tata Institute of Fundamental Research, Bombay, India
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.

Let X be a compact metric space with metric d. A complex-valued function ƒ on X is said to satisfy a Lipschitz condition if, for all points x and y of X, there exists a constant K such that

The smallest constant for which the above inequality holds is called the Lipschitz constant for ƒ and is denoted by ||ƒ||d, that is,

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 20 , 1968 , pp. 1150 - 1164
Copyright
Copyright © Canadian Mathematical Society 1968

Footnotes

The research in this paper forms part of the author's doctoral dissertation submitted to the Massachusetts Institute of Technology in June, 1966. The author would like to thank Professor K. Hoffman for guidance and encouragement in the course of this research. Thanks are due to the referee for pointing out some errors and for suggesting improvements of presentation in several places.

References

1. Dunford, Nelson and Schwartz, Jacob T., Linear operators. I. General theory (Interscience, New York, 1958).Google Scholar
2. Hoffman, K., Banach spaces of analytic functions (Prentice-Hall, Englewood Cliffs, N.J., 1962).Google Scholar
3. Kelley, J. L. and Namioka, Isaac, Linear topological spaces (Van Nostrand, Princeton, N.J., 1963).10.1007/978-3-662-41914-4CrossRefGoogle Scholar
4. de Leeuw, K., Banach spaces of Lipschitz functions, Studia Math. 21 (1961/62), 55–66.Google Scholar
5. Lindenstrauss, J., A short proof of Liapounoff's convexity theorem, J. Math. Mech. 15 1966), 971–972.Google Scholar
6. Phelps, R. R., Extreme points in function algebras, Duke Math. J. 32 (1965), 267–277.Google Scholar
7. Sherbert, Donald R., Banach algebras of Lipschitz functions, Pacific J. Math. 13 (1963), 1387-1399. g# Xhe structure of ideals and point derivations in Banach algebras of Lipschitz functions, Trans. Amer. Math. Soc. Ill (1964), 240–272.Google Scholar
9. Saks, Stanislaw, Theory of the integral, 2nd rev. ed. (Dover, New York, 1964).Google Scholar
10. Zygmund, A., Trigonometric series, 2nd éd., Vol. II (Cambridge Univ. Press, New York, 1959).Google Scholar