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

Question

Let $(V,|.|)$ be a normed finite dimensional vector space and $f:V\to V$ a map with the following property:

• $|f(y)|=|f(x+y)-f(x)|,\quad \forall x, y\in V.$

Then how to prove that $f$ is linear?

Update: what can be say if $V$ is a real vector space?

Answer 1

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 $f(z)=\overline z$.

Answer 2

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.