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A theorem of Stone-Weierstrass type

Published online by Cambridge University Press:  24 October 2008

G. A. Reid
G. A. Reid
Affiliation:
St John's College, Cambridge
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Extract

The Stone-Weierstrass theorem gives very simple necessary and sufficient conditions for a subset A of the algebra of all real-valued continuous functions on the compact Hausdorff space X to generate a subalgebra dense in namely, this is so if and only if the functions of A strongly separate the points of X, in other words given any two distinct points of X there exists a function in A taking different values at these points, and given any point of X there exists a function in A non-zero there. In the case of the algebra of all complex-valued continuous functions on X, the same result holds provided that we consider the subalgebra generated by A together with Ā, the set of complex conjugates of the functions in A.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 62 , Issue 4 , October 1966 , pp. 649 - 666
Copyright
Copyright © Cambridge Philosophical Society 1966

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References

REFERENCES

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