# 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:

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

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

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

iv) a complex non-separable infinite-dimensional Banach space is homeomorphic to its sphere?

Please answer even a part of my questions, that you know precisely.

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Crossposted on MO. –  Michael Greinecker Feb 16 at 8:33
Surely $f(x) = { x \over 1-\|x\|}$ is a homeomorphism between the open unit ball and its containing normed space (regardless of separability or real/complex)? –  copper.hat Feb 16 at 8:41
Sphere$=\{x:\|x\|=1\}$. –  Yiorgos S. Smyrlis Feb 16 at 9:39
I don't think the that the complex or real field makes any difference for this question... –  Henno Brandsma Feb 16 at 9:54

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 $X \smallsetminus \{0\}$.