Homeomorphisms between infinite-dimensional Banach spaces and their spheres As I know Cz. Bessaga has proved that an infinite-dimensional Banach space is homeomorphic to its unit sphere.
Unfortunately I do not have his book but I want to know is this theorem true without dependence from that the space is separable or not, and it is real or complex.
That is, is it true that:


*

*a real separable infinite-dimensional Banach space is homeomorphic to its sphere;

*a complex separable infinite-dimensional Banach space is homeomorphic to its sphere;

*a real non-separable infinite-dimensional Banach space is homeomorphic to its sphere;

*a complex non-separable infinite-dimensional Banach space is homeomorphic to its sphere?
 A: 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.
