For strictly convex spaces the result is true, and easy, even for isometries which are not necessarily surjective. In this case, for any and in , and for any in , write
and denote the closed ball of radius R around v by . Then is the unique element of , so, since is injective, is the unique element of
and therefore is equal to . Therefore is 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.
Jussi Väisälä (2003). "A Proof of the Mazur–Ulam Theorem". The American Mathematical Monthly. 110 (7): 633–635. doi:10.1080/00029890.2003.11920004. JSTOR3647749. S2CID 43171421.
mazur, ulam, theorem, mathematics, states, that, displaystyle, displaystyle, normed, spaces, over, mapping, displaystyle, colon, surjective, isometry, then, displaystyle, affine, proved, stanisław, mazur, stanisław, ulam, response, question, raised, stefan, ba. In mathematics the Mazur Ulam theorem states that if V displaystyle V and W displaystyle W are normed spaces over R and the mapping f V W displaystyle f colon V to W is a surjective isometry then f displaystyle f is affine It was proved by Stanislaw Mazur and Stanislaw Ulam in response to a question raised by Stefan Banach For strictly convex spaces the result is true and easy even for isometries which are not necessarily surjective In this case for any u displaystyle u and v displaystyle v in V displaystyle V and for any t displaystyle t in 0 1 displaystyle 0 1 writer u v V f u f v W displaystyle r u v V f u f v W and denote the closed ball of radius R around v by B v R displaystyle bar B v R Then t u 1 t v displaystyle tu 1 t v is the unique element of B v t r B u 1 t r displaystyle bar B v tr cap bar B u 1 t r so since f displaystyle f is injective f t u 1 t v displaystyle f tu 1 t v is the unique element of f B v t r B u 1 t r f B v t r f B u 1 t r B f v t r B f u 1 t r displaystyle f bigl bar B v tr cap bar B u 1 t r bigr f bigl bar B v tr bigr cap f bigl bar B u 1 t r bigr bar B bigl f v tr bigr cap bar B bigl f u 1 t r bigr and therefore is equal to t f u 1 t f v displaystyle tf u 1 t f v Therefore f displaystyle f is 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 See also editAleksandrov Rassias problemReferences editRichard J Fleming James E Jamison 2003 Isometries on Banach Spaces Function Spaces CRC Press p 6 ISBN 1 58488 040 6 Stanislaw Mazur Stanislaw Ulam 1932 Sur les transformations isometriques d espaces vectoriels normes C R Acad Sci Paris 194 946 948 Nica Bogdan 2012 The Mazur Ulam theorem Expositiones Mathematicae 30 4 397 398 arXiv 1306 2380 doi 10 1016 j exmath 2012 08 010 Jussi Vaisala 2003 A Proof of the Mazur Ulam Theorem The American Mathematical Monthly 110 7 633 635 doi 10 1080 00029890 2003 11920004 JSTOR 3647749 S2CID 43171421 nbsp This mathematical analysis related article is a stub You can help Wikipedia by expanding it vte nbsp This hyperbolic geometry related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Mazur Ulam theorem amp oldid 1169902092, wikipedia, wiki, book, books, library,
