Geodesic
On the Representation of Mappings of Compact Metrizable Spaces as Restrictions of Linear Transformations
Canadian journal of mathematics, Tome 22 (1970) no. 2, pp. 372-375

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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 l 2 exists such that hfh -1 is the restriction of a linear transformation. 
Edelstein, Michael. On the Representation of Mappings of Compact Metrizable Spaces as Restrictions of Linear Transformations. Canadian journal of mathematics, Tome 22 (1970) no. 2, pp. 372-375. doi: 10.4153/CJM-1970-045-1
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[1] 1. Edelstein, M., A short proof of a theorem by L. Janos, Proc. Amer. Math. Soc. 20 (1969), 509–510. Google Scholar

[2] 2. Edelstein, M., On the representation of contractive homeomorphisms as transformations in Hilbert space (to appear). Google Scholar

[3] 3. Janos, L., A converse of Banach’s contraction theorem, Proc. Amer. Math. Soc. 18 (1967), 287–289. Google Scholar

[4] 4. Janos, L., Linearization of a contractive homeomorphism, Can. J. Math. 20 (1968), 1387–1390. Google Scholar

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