Mazur–Ulam theorem
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Mazur–Ulam theorem
Mazur–Ulam theorem
In mathematics, the Mazur–Ulam theorem states that ifandarenormed spacesover R and themapping
is a surjectiveisometry, thenisaffine.
It is named afterStanisław MazurandStanisław Ulamin response to an issue raised byStefan Banach.
Forstrictly convex spacesthe result is true, and easy, even for isometries which are not necessarily surjective. In this case, for anyandin, and for anyin, denoting, one has thatis the unique element of, so, beinginjective,is the unique element of, namely. Thereforeis an affine map. This argument fails in the general case, because in a normed space which is not strictly convex two tangent balls may meet in some flat convex region of their boundary, not just a single point.
![[0,1]](https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d)
References
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Sep 29, 2019, 3:19 AM