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A remark on a theorem of Caradus

Published online by Cambridge University Press:  17 April 2009

J.A. Johnson
J.A. Johnson
Affiliation:
Oklahoma State University, Stillwater, Oklahoma, USA.
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It is shown how a result of S.R. Caradus on the approximation problem can be obtained from known theorems.

Terms used here are standard (see [1] or [3]).

Let X denote a Banach space, S its unit ball in the weak topology, and X* the dual of X. In [1], the following theorem is proved: (I) If X is reflexive and X* (considered as a subspaoe of the continuous scalar-valued functions C(S) in the canonical way) is complemented in C(S), then X has the approximation property.

It is our purpose to point out that (I) is a corollary to some known theorems that yield the stronger conclusion (II) below.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 6 , Issue 3 , June 1972 , pp. 355 - 356
Copyright
Copyright © Australian Mathematical Society 1972

References

[1]Caradus, S.R., “The approximation problem for compact operators”, Bull. Austral. Math. Soc. 1 (1969), 397–401.CrossRefGoogle Scholar
[2]de Lamadrid, Jesús Gil, “On finite dimensional approximations of mappings in Banach spaces”, Proc. Amer. Math. Soc. 13 (1962), 163–168.CrossRefGoogle Scholar
[3]Schaefer, Helmut H., Topologiaal vector spaces (The Macmillan Company, New York; Collier-Macmillan, London; 1966).Google Scholar