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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:

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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    $\begingroup$ Crossposted on MSE. $\endgroup$
    – Michael Greinecker
    Commented Feb 16, 2014 at 8:32
  • $\begingroup$ You do not need to bother about $\mathbb{C}$. A complex Banach space is a particular case of a real Banach space by restriction of the scalar field, and a homeomorphism is just about topology. $\endgroup$
    – Pietro Majer
    Commented Feb 16, 2014 at 8:54
  • $\begingroup$ Not an answer but some references that might be helpful: Bessaga and Pelczynski penned a survey "Selected topics in infinite dimensional topology" and their work was continued to the non-separable case by H. Torunczyk and then by T. Banakh. The latter's articles are available on arXiv. $\endgroup$
    – alpha
    Commented Feb 16, 2014 at 10:21

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