Mazur–Ulam theorem

Mazur–Ulam theorem In mathematics, the Mazur–Ulam theorem states that if {displaystyle V} and {displaystyle W} are normed spaces over R and the mapping {displaystyle fcolon Vto W} is a surjective isometry, then {displaystyle f} is affine.

It is named after Stanisław Mazur and Stanisław Ulam in response to an issue 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 {displaystyle u} and {displaystyle v} in {displaystyle V} , and for any {displaystyle t} in {displaystyle [0,1]} , denoting {displaystyle r:=|u-v|_{V}=|f(u)-f(v)|_{W}} , one has that {displaystyle tu+(1-t)v} is the unique element of {displaystyle {bar {B}}(v,tr)cap {bar {B}}(u,(1-t)r)} , so, being {displaystyle f} injective, {displaystyle f(tu+(1-t)v)} is the unique element of {displaystyle f{big (}{bar {B}}(v,tr)cap {bar {B}}(u,(1-t)r{big )}=f{big (}{bar {B}}(v,tr){big )}cap f{big (}{bar {B}}(u,(1-t)r{big )}={bar {B}}{big (}f(v),tr{big )}cap {bar {B}}{big (}f(u),(1-t)r{big )}} , namely {displaystyle tf(u)+(1-t)f(v)} . Therefore {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.

References Richard J. Fleming; James E. Jamison (2003). Isometries on Banach Spaces: Function Spaces. CRC Press. p. 6. ISBN 1-58488-040-6. Stanisław Mazur; Stanisław Ulam (1932). "Sur les transformations isométriques d'espaces vectoriels normés". C. R. Acad. Sci. Paris. 194: 946–948. Jussi Väisälä (2003). "A Proof of the Mazur-Ulam Theorem". The American Mathematical Monthly. 110 (7): 633–635. doi:10.1080/00029890.2003.11920004. S2CID 43171421. External links Nica, Bogdan (2013). "A proof of the Mazur–Ulam theorem assuming f is bijective". arXiv:1306.2380. Väisälä, Jussi. "A proof of the Mazur–Ulam theorem" (PDF). Archived from the original (PDF) on 16 May 2018. show vte Banach space topics show vte Functional analysis (topics – glossary) This mathematical analysis–related article is a stub. You can help Wikipedia by expanding it.

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