Homeomorphisms between infinite-dimensional Banach spaces and their spheres

Bessaga showed something stronger, but only for Hilbert spaces. Generalization to certain Banach spaces (i.e., those which are linearly injectable into some $c_0(\Gamma)$) was given by Dobrowolski. The following paragraph is from Diffeomorphisms between spheres and hyperplanes in infinite-dimensional Banach spaces by D. Azagra, Studia Math. 125 (1997), no. 2, 179–186.

In 1966 C. Bessaga [1] proved that every infinite-dimensional Hilbert space $H$ is $C^\infty$ diffeomorphic to its unit sphere. The key to prove this astonishing result was the construction of a diffeomorphism between $H$ and $H \smallsetminus \{0\}$ being the identity outside a ball, and this construction was possible thanks to the existence of a $C^\infty$ non-complete norm in $H$. In 1979 T. Dobrowolski [2] developed Bessaga’s non-complete norm technique and proved that every infinite-dimensional Banach space $X$ which is linearly injectable into some $c_0(\Gamma)$ is $C^\infty$ diffeomorphic to $X \smallsetminus \{0\}$.

[1] Bessaga, C. Every infinite-dimensional Hilbert space is diffeomorphic with its unit sphere. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 14 (1966), 27–31.

[2] Dobrowolski, T., Smooth and R-analytic negligibility of subsets and extension of homeomorphism in Banach spaces, Studia Math. 65 (1979), 115-139.