Mazur–Ulam theorem

In mathematics, the Mazur–Ulam theorem states that if [math]\displaystyle{ V }[/math] and [math]\displaystyle{ W }[/math] are normed spaces over R and the mapping

[math]\displaystyle{ f\colon V\to W }[/math]

is a surjective isometry, then [math]\displaystyle{ f }[/math] is affine. It was proved by Stanisław Mazur and Stanisław 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 [math]\displaystyle{ u }[/math] and [math]\displaystyle{ v }[/math] in [math]\displaystyle{ V }[/math], and for any [math]\displaystyle{ t }[/math] in [math]\displaystyle{ [0,1] }[/math], write [math]\displaystyle{ r=\|u-v\|_V=\|f(u)-f(v)\|_W }[/math] and denote the closed ball of radius R around v by [math]\displaystyle{ \bar B(v,R) }[/math]. Then [math]\displaystyle{ tu+(1-t)v }[/math] is the unique element of [math]\displaystyle{ \bar B(v,tr)\cap \bar B(u,(1-t)r) }[/math], so, since [math]\displaystyle{ f }[/math] is injective, [math]\displaystyle{ f(tu+(1-t)v) }[/math] is the unique element of [math]\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), }[/math] and therefore is equal to [math]\displaystyle{ tf(u)+(1-t)f(v) }[/math]. Therefore [math]\displaystyle{ f }[/math] 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

Aleksandrov–Rassias problem

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. https://gallica.bnf.fr/ark:/12148/bpt6k31473/f950.item. 
• Nica, Bogdan (2012). "The Mazur–Ulam theorem". Expositiones Mathematicae 30 (4): 397–398. doi:10.1016/j.exmath.2012.08.010. 
• Jussi Väisälä (2003). "A Proof of the Mazur–Ulam Theorem". The American Mathematical Monthly 110 (7): 633–635. doi:10.1080/00029890.2003.11920004.