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On the Representation of Mappings of Compact Metrizable Spaces as Restrictions of Linear Transformations

Published online by Cambridge University Press:  20 November 2018

Michael Edelstein
Michael Edelstein*
Affiliation:
Dalhousie University, Halifax, Nova Scotia
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Let f: X → X be a continuous mapping of the compact metrizable space X into itself with a singleton. In [3] Janos proved that for any λ, 0 < λ < 1, a metric ρ compatible with the topology of X exists such that ρ(f(x), f(y)) ≦ λρ(x, y) for all x, y ∈X. More recently, Janos [4] has shown that if, in addition, f is one-to-one, then a Hilbert space H and a homeomorphism μ: X → H exist such that μfμ-1 is the restriction to μ[X] of the transformation sending y ∈ H into λy. Our aim in this note is to show that in both cases a homeomorphism h of X into l2 exists such that hfh-1 is the restriction of a linear transformation.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 2 , 01 April 1970 , pp. 372 - 375
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Edelstein, M., A short proof of a theorem by L. Janos, Proc. Amer. Math. Soc. 20 (1969), 509–510.Google Scholar
2. Edelstein, M., On the representation of contractive homeomorphisms as transformations in Hilbert space (to appear).Google Scholar
3. Janos, L., A converse of Banach’s contraction theorem, Proc. Amer. Math. Soc. 18 (1967), 287–289.Google Scholar
4. Janos, L., Linearization of a contractive homeomorphism, Can. J. Math. 20 (1968), 1387–1390.Google Scholar