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Every infinite-dimensional Hilbert space is real-analytically isomorphic with its unit sphere. (English) Zbl 0869.46013
Let $${\mathfrak E}= ({\mathfrak E},|\cdot|)$$ be a separable Banach space, $$\omega$$ a real analytic (resp., $${\mathfrak C}^p$$, $$p=1,2,\dots,\infty$$) norm on $${\mathfrak E}$$, and $${\mathfrak E}_0=\ker x^*$$ for a $$x^*\neq 0$$ in $${\mathfrak E}^*$$. The author proves that $${\mathfrak S}_\omega=\{x\in {\mathfrak E}\mid\omega(x)=1\}$$ and $${\mathfrak E}_0$$ are real-analytically (resp., $${\mathfrak C}^p$$) isomorphic. His result derives the following Theorem:
Let $$\mathfrak H$$ be an infinite-dimensional Hilbert space, and $$\mathfrak S$$ the unit sphere in $$\mathfrak H$$. There exists a real-analytic isomorphism $$f$$ from $$\mathfrak S$$ onto $$\mathfrak H$$.
His Theorem provides an affirmative answer to the problem by C. Bessaga [Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 14, 27-31 (1966; Zbl 0151.17703)]. Next let $$K$$ be a compact set in $$\mathfrak S$$. As a final Remark, he obtains a $${\mathfrak C}^p$$ isomorphism $$f:{\mathfrak E}\backslash K\to{\mathfrak E}$$ in the infinite-dimensional separable Banach space $$\mathfrak E$$ with $${\mathfrak C}^p$$ norm.
Reviewer: H.Yamagata (Osaka)

##### MSC:
 46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product) 46B03 Isomorphic theory (including renorming) of Banach spaces
##### Keywords:
real-analytic isomorphism
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