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Diffeomorphisms between spheres and hyperplanes in infinite-dimensional Banach spaces. (English) Zbl 0899.58007
The paper is centered around the following main theorem.
Theorem 1. Let \((X,\|\cdot\|)\) be an infinite-dimensional Banach space with a \(C^p\) smooth norm \(\|\cdot\|\), and let \(S_X\) be the unit sphere. Then, for every closed hyperplane \(H\) in \(X\), there exists a \(C^p\) diffeomorphism between \(S_X\) and \(H\).
The proof is based on a previous result of C. Bessaga [Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 14, 27-31 (1966; Zbl 0151.17703)] (every infinite-dimensional Hilbert space \(H\) is \(C^\infty\) diffeomorphic to its unit sphere), the Banach contraction mapping principle and the following fixed point lemma: let \(F:(0,\infty)\to [0,\infty)\) be a continuous function such that, for every \(\beta\geq\alpha> 0\), \[ F(\beta)- F(\alpha)\leq\textstyle{{1\over 2}} (\beta- \alpha)\quad\text{and}\quad \displaystyle{{\limsup_{t\to 0^+}}} F(t)> 0. \] Then there exists a unique \(\alpha> 0\) such that \(F(\alpha)= \alpha\).

58B99 Infinite-dimensional manifolds
46B20 Geometry and structure of normed linear spaces
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