# zbMATH — the first resource for mathematics

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 $$F(\alpha)= \alpha$$.

##### MSC:
 58B99 Infinite-dimensional manifolds 46B20 Geometry and structure of normed linear spaces
Full Text: