Let $\ell_2:=\{x=(x_n)_{n\in\mathbb N}:\ \|x\|^2:=\sum_n|x_n|^2<\infty\}$ with its natural norm. According to Wikipedia https://en.wikipedia.org/wiki/Kuiper%27s_theorem and to other sources, it is well-known that there exists an invertible map $\phi: \ell_2\setminus\{0\} \to \ell_2$, with different degrees of regularity (homeomorphism, or $C^\infty$ or analytic, can all be done, Google points out), and which satisfies $\phi(x)=x$ for $\|x\|\ge 1$. I would like to know more about the $C^\infty$ case.
The initial idea seems to be due to Bessaga, in his paper "Every infinite dimensional Hilbert space is diffeomorphic with its unit sphere", Bull. Acad. Polon. Sci., 14 (1966), 27-31.
One thing that even I know, is that $x=(x_1,x_2,\ldots)\mapsto (\sqrt{1-\|x\|^2}, x_1,x_2,\ldots)$ gives a nice map from the unit ball to the unit sphere in $\ell_2$, but I'm not sure if this helps towards the above "Bessaga theorem", and neither was I able find Bessaga's paper, or to find the right functional analysis book to look at.
I'd appreciate pointer to either (a) the ideas for the proof, (b) a PDF of Bessaga's paper, or (c) a reference of a book that can help.
