| Version 6 |
Version 5 |
| \PMlinkescape{isometric} |
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| \PMlinkescape{isometries} |
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| \PMlinkescape{isometry} |
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| \begin{thm*} |
\begin{thm*} |
| Every \PMlinkname{isometry}{Isometry} between \PMlinkname{normed vector spaces}{NormedVectorSpace} over $\R$ |
Every \PMlinkname{isometry}{Isometry} between \PMlinkname{normed vector spaces}{NormedVectorSpace} over $\R$ |
| is an affine transformation. |
is an affine transformation. |
| \end{thm*} |
\end{thm*} |
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| Note that we consider isometries to be surjective by definition. |
This result was first proved by Mazur and Ulam.\cite{mazurulam} |
| The result is not in general true for non-surjective isometric mappings. |
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| This theorem was first proved by Mazur and Ulam.\cite{mazurulam} |
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| A simpler proof has been given by Jussi V\"{a}is\"{a}l\"{a}.\cite{vaisala} |
A simpler proof has been given by Jussi V\"{a}is\"{a}l\"{a}.\cite{vaisala} |
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| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{mazurulam} |
\bibitem{mazurulam} |
| S.\ Mazur and S.\ Ulam, |
S.\ Mazur and S.\ Ulam, |
| {\it Sur les transformations isom\'etriques d'espaces vectoriels norm\'es}, |
{\it Sur les transformations isom\'etriques d'espaces vectoriels norm\'es}, |
| C.\ R.\ Acad.\ Sci., Paris 194 (1932), 946--948. |
C.\ R.\ Acad.\ Sci., Paris 194 (1932), 946--948. |
| \bibitem{vaisala} |
\bibitem{vaisala} |
| Jussi V\"{a}is\"{a}l\"{a}, |
Jussi V\"{a}is\"{a}l\"{a}, |
| {\it A proof of the Mazur--Ulam theorem}, |
{\it A proof of the Mazur--Ulam theorem}, |
| Amer.\ Math.\ Mon.\ 110, \#7 (2003), 633--635. |
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}.) |
(A preprint is \PMlinkexternal{available on V\"{a}is\"{a}l\"{a}'s website}{http://www.helsinki.fi/\%7Ejvaisala/mazurulam.pdf}.) |
| \end{thebibliography} |
\end{thebibliography} |
