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A remark on a theorem of Caradus
Published online by Cambridge University Press: 17 April 2009
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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.
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- Research Article
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- 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
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