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A counterexample to the density of the space generated by the composition operators

Published online by Cambridge University Press:  17 April 2009

Ronald Beattie
Ronald Beattie
Affiliation:
Lehrstuhl für Mathematik I, Universität Mannheim, Mannheim, Federal Republic of Germany.
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Abstract

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It is known that, for an arbitrary convergence space X, the vector space generated by X is dense in LcCc (X) where both C(X) and its dual space carry the continuous convergence structure. In this note, a natural analogue formulated for the operator space L(Cc(X), Cc(X)) is considered, namely: is the vector space generated by the composition operators associated to the continuous mappings in C(X, X) dense in Lc (Cc (X), Cc (X)) ? This question is answered in the negative by a counterexample.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 16 , Issue 1 , February 1977 , pp. 79 - 81
Copyright
Copyright © Australian Mathematical Society 1977

References

[1]Binz, E., Continuous convergence on C(X) (Lecture Notes in Mathematics, 469. Springer-Verlag, Berlin, Heidelberg, New York, 1975).CrossRefGoogle Scholar
[2]Butzmann, H.-P., “Über die c–Reflexivität von Cc (X)”, Comment. Math. Helv. 47 (1972), 92–101.CrossRefGoogle Scholar