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
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If you assume $f$ to be surjective then $f$ has to be linear without any assumptions on $V$ by the MazurUlam theorem. Wikipedia doesn't offer much more information than a link to the beautiful recent proof by J. Väisälä. 

