When do 0-preserving isometries have to be linear? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T14:11:59Z http://mathoverflow.net/feeds/question/62380 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/62380/when-do-0-preserving-isometries-have-to-be-linear When do 0-preserving isometries have to be linear? Ricky Demer 2011-04-20T06:58:36Z 2011-04-20T07:16:01Z <p>Let $\langle \mathbf{V},+,\cdot,||.|| \rangle$ be a normed vector space over $\mathbb{R}$. <br> Let $f : \mathbf{V} \to \mathbf{V}$ be an isometry that satisfies $f(\mathbf{0}) = \mathbf{0}$ . <br> What conditions on the vector space would or would not force $f$ to be linear? <br><br> 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#62382 Answer by Theo Buehler for When do 0-preserving isometries have to be linear? Theo Buehler 2011-04-20T07:16:01Z 2011-04-20T07:16:01Z <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.&nbsp;V&auml;is&auml;l&auml;.</p>