Hostname: page-component-77c89778f8-gq7q9 Total loading time: 0 Render date: 2024-07-24T04:58:36.523Z Has data issue: false hasContentIssue false
  • English
  • Français

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
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.

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

REFERENCES

1.Alfsen, E. and Effros, E., Structure in real Banach spaces, Ann. Math. 96 (1972), 98–173.Google Scholar
2.Blatter, J., Grothendieck Spaces in Approximation Theory, Amer. Math. Soc. Memoir 120 (Providence, R.I., 1972).Google Scholar
3.Dieudonné, J., Une généralisation des espaces compacts, j. Math. Pures Appl. 23 (1944), 65–76.Google Scholar
4.Fakhoury, H., Projections de meilleure approximation continues dans certains espaces de Banach, C.R. Acad. Sci. Paris 276 (1973), A45–A48.Google Scholar
5.Holmes, R., A Course on Optimization and Best Approximation (Springer-Verlag, Berlin-Heidelberg-New York, 1972).CrossRefGoogle Scholar
6.Holmes, R. and Kripke, B., Approximation of bounded functions by continuous functions, Bull. Amer. Math. Soc. 71 (1965), 896–897.Google Scholar
7.Holmes, R. and Kripke, B., Interposition and approximation, Pacific J. Math. 24 (1968), 103–110.Google Scholar
8.Holmes, R. and Kripke, B., Best approximation by compact operators, Indiana Univ. Math. J. 21 (1971), 255–263.CrossRefGoogle Scholar
9.Olech, C., Approximation of set-valued functions by continuous functions, Collect. Math. 19 (1968), 285–293.Google Scholar
10.Sakai, S., C*-Algebras and W*-Algebras (Springer-Verlag, New York-Heidelberg-Berlin, 1971).Google Scholar