When do 0-preserving isometries have to be linear? - MathOverflow most recent 30 from http://mathoverflow.net2013-05-24T14:11:59Zhttp://mathoverflow.net/feeds/question/62380http://www.creativecommons.org/licenses/by-nc/2.5/rdfhttp://mathoverflow.net/questions/62380/when-do-0-preserving-isometries-have-to-be-linearWhen do 0-preserving isometries have to be linear?Ricky Demer2011-04-20T06:58:36Z2011-04-20T07:16:01Z
<p>Let $\langle \mathbf{V},+,\cdot,||.|| \rangle$ be a normed vector space over $\mathbb{R}$.
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Let $f : \mathbf{V} \to \mathbf{V}$ be an isometry that satisfies $f(\mathbf{0}) = \mathbf{0}$ .
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What conditions on the vector space would or would not force $f$ to be linear?
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examples: finite dimensional, complete, norm induced by an inner product, strictly convex</p>
http://mathoverflow.net/questions/62380/when-do-0-preserving-isometries-have-to-be-linear/62382#62382Answer by Theo Buehler for When do 0-preserving isometries have to be linear?Theo Buehler2011-04-20T07:16:01Z2011-04-20T07:16:01Z<p>If you assume $f$ to be surjective then $f$ has to be linear without any assumptions on $V$ by the <a href="http://en.wikipedia.org/wiki/Mazur%2DUlam_theorem" rel="nofollow">Mazur-Ulam theorem</a>. Wikipedia doesn't offer much more information than a link to the beautiful recent <a href="http://www.helsinki.fi/~jvaisala/mazurulam.pdf" rel="nofollow">proof</a> by J. Väisälä.</p>