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An approximative property of spaces of continuous functions

Published online by Cambridge University Press:  18 May 2009

R. B. Holmes  and
J. D. Ward
R. B. Holmes
Affiliation:
Purdue University, West Lafayette, Indiana 47907
J. D. Ward
Affiliation:
Purdue University, West Lafayette, Indiana 47907
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A Banach space X is said to have property (PROXBID) if the canonical image of X in its bidual X** is proximal. In other words, if J: X → X** is the canonical embedding, then it is required that every element of X** have at least one best approximation (i.e., nearest point) from the closed subspace J(X). We show below that, if X is the space of (real or complex) continuous functions on a compact set, or the space of (real or complex) continuous functions that vanish at infinity on a locally compact set, then X has property (PROXBID). At this point we should mention the existence of a variety of examples [2, 8] of Banach spaces which lack property (PROXBID).

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 15 , Issue 1 , March 1974 , pp. 48 - 53
Copyright
Copyright © Glasgow Mathematical Journal Trust 1974

References

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