# 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.

#### Solutions Collecting From Web of "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.