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Every bijective isometry between normed spaces is affine. This well-known and beautiful statement, the Mazur-Ulam Theorem, was proved in 1932, but the proof has been simplified and polished in years, by contributions of more people (Lyusternik, Borsuk, Vogt, Väisälä, and others I'm not aware of) until the final form of a little gem (the last version I know, by Bogdan Nica, can be explained in conversation).

Reaching a linearity property from a metric assumption is something peculiar. Another result of this "from metric to linear" form, that comes to my mind, is the linearity of the metric projector on a subspace of a Hilbert space. However, the latter is a fundamental result with very important and rich consequences on the structure of Hilbert spaces, and applications in the Calculus of Variations. On the opposite, I know no application of Mazur-Ulam theorem, and I would be really glad to learn some. Applications of any generalization of it are also welcome.

edit. Of course, knowing that surjective isometries on a normed space are linear has a clear foundational application, since any theory about isometries of normed spaces can assume linearity, and exploit the linear theory of operators. Then the question is: what natural sources of (a priori non linear) isometries of normed spaces are there?

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    $\begingroup$ One of my recent papers used an extension due to Dilworth: arxiv.org/abs/1610.02381 $\endgroup$
    – Anthony Quas
    Commented Sep 27, 2019 at 15:29
  • $\begingroup$ If $E$ is a Banach space and $G$ is group acting on $E$ by linear isometries with no nonzero fixed vector, but with almost fixed vectors (that is for every $n$ and finite subset of the group $G$, there exists a unit vector moved by $\le 1/n$ by each element of the subset). Then one can produce some fixed-point-free limit action by zooming on the sphere (at a well-chosen sequence of points and choose an ultrafilter). The limit action is clearly isometric, but the limit space is not even a normed space in general; still it is Banach if $E$ is smooth. $\endgroup$
    – YCor
    Commented Oct 14, 2019 at 17:01

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The Mazur-Ulam theorem is used in Bader, Uri; Furman, Alex; Gelander, Tsachik; Monod, Nicolas Property (T) and rigidity for actions on Banach spaces. Acta Math. 198 (2007), no. 1, 57–105 in order to simplify the theory of isometric actions on Banach spaces. In this connection it is worth mentioning that the question on an analogue of the Mazur-Ulam theorem for convex sets (which they ask on page 88) was answered by Schechtman, see Generalizing the Mazur-Ulam theorem to convex sets with empty interior in Banach spaces

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  • $\begingroup$ This is interesting, but could you add some details about how the theorem simplifies the theory of isometric actions on Banach spaces (a part from the fact that the theory is immediately reduced to the theory of linear isometric actions). I added a remark about this. Thank you $\endgroup$
    – Pietro Majer
    Commented Oct 14, 2019 at 16:28
  • $\begingroup$ I think that it is the main application of such rigidity results (a weak condition implies a stronger one) - you do not have to develop the theory of objects which satisfy the weak condition, but not the strong one. $\endgroup$
    – August Cleaner
    Commented Oct 15, 2019 at 16:21

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