Let $\langle \mathbf{V},+,\cdot,||.|| \rangle$ be a normed vector space over $\mathbb{R}$.
  

Let $f : \mathbf{V} \to \mathbf{V}$ be an isometry that satisfies $f(\mathbf{0}) = \mathbf{0}$ .
  

What conditions on the vector space would or would not force $f$ to be linear?
  
  

examples: finite dimensional, complete, norm induced by an inner product, strictly convex



When do 0-preserving isometries have to be linear?
If you assume $f$ to be surjective then $f$ has to be linear without any assumptions on $V$ by the [Mazur-Ulam theorem](http://en.wikipedia.org/wiki/Mazur%2DUlam_theorem). Wikipedia doesn't offer much more information than a link to the beautiful recent [proof](https://web.archive.org/web/20180516125105/http://www.helsinki.fi/~jvaisala/mazurulam.pdf) by J. Väisälä.


