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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?

Xander Henderson
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dankan257
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1 Answers1

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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.

Yiorgos S. Smyrlis
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  • Reading this, I am wondering another related question: does there exist a (Nash-type) isometric embedding theorem for Hilbert manifolds in its model Hilbert space? There are some works of the '60s building $C^{\infty}$ embeddings, and Dobrowolski proved in 1995 that this can be made even real analytic. But my guess is that there does not exist such an object. Have you ever hear of something related? – Crash Bandicoot Feb 14 '24 at 15:41