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Suppose that $X$ and $Y$ are two Banach spaces, $S_{X}$ and $S_{Y}$ their unit spheres, and $f$ an onto isometry between $S_X$ and $S_Y$. Does it follow that $X$ and $Y$ are isometric?

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    $\begingroup$ I believe this is open when stated in full-generality, being sometimes referred to as Tingley's problem. I do not know the details here, but perhaps having this name to search for may help you out; for instance, the answer is apparently positive if $X$ or $Y$ is one of the classical sequence spaces $\endgroup$
    – Yemon Choi
    Mar 24, 2020 at 1:39
  • $\begingroup$ (I am assuming you are taking real scalars everywhere) $\endgroup$
    – Yemon Choi
    Mar 24, 2020 at 1:39
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    $\begingroup$ As of November 2018, the problem seems to be still open arxiv.org/pdf/1804.10674.pdf $\endgroup$ Mar 24, 2020 at 7:39
  • $\begingroup$ Is there a reason why the insistence on real scalars? Doesn't the problem make sense for complex too? $\endgroup$
    – Markus
    Mar 24, 2020 at 22:36
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    $\begingroup$ @Markus: There are many examples of complex Banach spaces that are not isomorphic as complex Banach spaces but are isometrically isomorphic as real Banach spaces. See MR0818448. $\endgroup$ Mar 26, 2020 at 2:08

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