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By a result of Godefroy and Kalton if $X,Y$ are separable Banach spaces and $X$ embeds isometrically into $Y$, then $X$ embeds with a linear isometry into $Y$.

Is this result known to fail for nonseparable spaces? That is, is there a known example of two (necessarily nonseparable) Banach spaces $X,Y$ such that $X$ embeds isometrically into $Y$, but such that there is no linear isometric embedding of $X$ into $Y$?

This question was crossposted to MO and answered there.

TopologicalDynamitard
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  • Why do you think such an example exists? Or might even be “simple”? Within $C_p$ theory this is a well studied problem ( but that “ merely” gives a TVS (locally convex) as examples. – Henno Brandsma Jul 07 '21 at 21:42
  • @HennoBrandsma that's a good point, I am expecting this result to fail for nonseparable spaces, but I don't know for sure that it does. I'm updating the question to reflect this. – TopologicalDynamitard Jul 08 '21 at 09:23

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This question was answered by Nik Weaver on mathoverflow in the positive: indeed this result is known to be false for nonseparable spaces.

GEdgar
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TopologicalDynamitard
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  • This is (almost) how it should be done. If not answered here after 5 months, then ask in MO. Actually, one week should be enough delay. – GEdgar Dec 06 '21 at 13:37