Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-26T01:13:09.288Z Has data issue: false hasContentIssue false
  • English
  • Français

Approximation and extension of continuous functions

Part of: Linear function spaces and their duals Maps and general types of spaces defined by maps Fairly general properties

Published online by Cambridge University Press:  09 April 2009

Salvador Hernández-Muñoz
Rights & Permissions [Opens in a new window]

Abstract

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.

In this paper we study the approximation of vector valued continuous functions defined on a topological space and we apply this study to different problems. Thus we give a new proof of Machado's Theorem. Also we get a short proof of a Theorem of Katětov and we prove a generalization of Tietze's Extension Theorem for vector-valued continuous functions, thereby solving a question left open by Blair.

MSC classification

Secondary: 54C20: Extension of maps 54C30: Real-valued functions 54D60: Realcompactness and realcompactification 46E25: Rings and algebras of continuous, differentiable or analytic functions 54C45: $C$- and $C^*$-embedding 54C99: None of the above, but in this section 46E15: Banach spaces of continuous, differentiable or analytic functions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 57 , Issue 2 , October 1994 , pp. 149 - 157
Copyright
Copyright © Australian Mathematical Society 1994

References

[1]Alo, R. A. and Shapiro, H. L., Normal topological spaces (Cambridge University Press, London, 1974).Google Scholar
[2]Bishop, E., ‘A generalization of the Stone-Weierstrass Theorem’, Pacific J. Math. 11 (1961), 777–783.CrossRefGoogle Scholar
[3]Blair, R. L., ‘Extensions of Lebesgue sets and of real-valued functions’, Czechoslovak Math. J. 31 (1981), 63–74.CrossRefGoogle Scholar
[4]Burckel, R. B., ‘Bishop's Stone-Weierstrass Theorem’, Amer. Math. Monthly 91 (1984), 22–32.Google Scholar
[5]Edwards, D. A., ‘A short proof of a Theorem of Machado’, Math. Proc. Cambridge Philos. Soc. 99 (1986), 111–114.CrossRefGoogle Scholar
[6]Gillman, L. and Jerison, M., Rings of continuous functions (Van Nostrand, Princeton, 1960).CrossRefGoogle Scholar
[7]Katětov, M., ‘Measures in fully normal spaces’, Fund. Math. 38 (1951), 73–84.CrossRefGoogle Scholar
[8]Machado, S., ‘On Bishop's generalization of the Stone-Weierstrass Theorem’, Indag. Math. 39 (1977), 218–224.CrossRefGoogle Scholar
[9]Mrówka, S., ‘On some approximation theorems’, Nieuw Arch. Wisk. 16 (1968), 94–111.Google Scholar
[10]Ransford, T. J., ‘A short elementary proof of the Stone-Weierstrass-Bishop Theorem’, Math. Proc. Cambridge Philos. Soc. 96 (1984), 309–311.CrossRefGoogle Scholar