TITLE: When do 0-preserving isometries have to be linear?
QUESTION [6 upvotes]: 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

REPLY [14 votes]: If you assume $f$ to be surjective then $f$ has to be linear without any assumptions on $V$ by the Mazur-Ulam theorem. Wikipedia doesn't offer much more information than a link to the beautiful recent proof by J. Väisälä.