Azagra, Daniel Diffeomorphisms between spheres and hyperplanes in infinite-dimensional Banach spaces. (English) Zbl 0899.58007 Stud. Math. 125, No. 2, 179-186 (1997). 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 \(F(\alpha)= \alpha\). Reviewer: V.G.Angelov (Sofia) Cited in 9 Documents MSC: 58B99 Infinite-dimensional manifolds 46B20 Geometry and structure of normed linear spaces Keywords:infinite-dimensional Banach space; unit sphere; hyperplane; diffeomorphism Citations:Zbl 0151.17703 PDFBibTeX XMLCite \textit{D. Azagra}, Stud. Math. 125, No. 2, 179--186 (1997; Zbl 0899.58007) Full Text: DOI EuDML