Mazur-Ulam theorem
MazurUlamTheorem
2006-11-05 10:25:17
2006-11-25 05:51:32
Theorem
normed vector space
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\newtheorem*{thm*}{Theorem}
\def\C{\mathbb{C}}
\def\R{\mathbb{R}}
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\PMlinkescapeword{isometric}
\PMlinkescapeword{isometries}
\PMlinkescapeword{isometry}
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\begin{thm*}
Every \PMlinkname{isometry}{Isometry} between normed vector spaces over $\R$
is an affine transformation.
\end{thm*}
Note that we consider isometries to be surjective by definition.
The result is not in general true for non-surjective isometric mappings.
The result does not extend to normed vector spaces over $\C$,
as can be seen from the fact that complex conjugation is an isometry $\C\to\C$
but is not affine over $\C$.
(But complex conjugation is clearly affine over $\R$,
and in general any normed vector space over $\C$
can be considered as a normed vector space over $\R$,
to which the theorem can be applied.)
This theorem was first proved by Mazur and Ulam.\cite{mazurulam}
A simpler proof has been given by Jussi V\"{a}is\"{a}l\"{a}.\cite{vaisala}
\begin{thebibliography}{9}
\bibitem{mazurulam}
S.\ Mazur and S.\ Ulam,
{\it Sur les transformations isom\'etriques d'espaces vectoriels norm\'es},
C.\ R.\ Acad.\ Sci., Paris 194 (1932), 946--948.
\bibitem{vaisala}
Jussi V\"{a}is\"{a}l\"{a},
{\it A proof of the Mazur--Ulam theorem},
Amer.\ Math.\ Mon.\ 110, \#7 (2003), 633--635.
(A preprint is \PMlinkexternal{available on V\"{a}is\"{a}l\"{a}'s website}{http://www.helsinki.fi/\%7Ejvaisala/mazurulam.pdf}.)
\end{thebibliography}