# 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:

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

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

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

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

• Crossposted on MO. – Michael Greinecker Feb 16 '14 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 '14 at 8:41
• Sphere$=\{x:\|x\|=1\}$. – Yiorgos S. Smyrlis Feb 16 '14 at 9:39
• I don't think the that the complex or real field makes any difference for this question... – Henno Brandsma Feb 16 '14 at 9:54

## 1 Answer

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  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  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 $X \smallsetminus \{0\}$.

 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.

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