How to show that $f:V\to V$ is linear?

This is false. Let $f:\mathbb R \to \mathbb R$ be an additive function. Then the hypothesis is satisfied. But without some continuity/measurability assumption we cannot conclude that $f$ is linear.

Without specifying the base field this is false even if we interprete "map" as continuous: Consider $\Bbb C$ with standard norm as one-dimensional complex vector space and let $f(z)=\overline z$.