# Isometries on the unit sphere

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?

• 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 Mar 24, 2020 at 1:39
• (I am assuming you are taking real scalars everywhere) Mar 24, 2020 at 1:39
• As of November 2018, the problem seems to be still open arxiv.org/pdf/1804.10674.pdf Mar 24, 2020 at 7:39
• Is there a reason why the insistence on real scalars? Doesn't the problem make sense for complex too? Mar 24, 2020 at 22:36
• @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. Mar 26, 2020 at 2:08