Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 22F / c / Solution

Assume $u(0)=0$. Applying midpoint preservation to $2v$ and $0$ givesso $u(2v)=2u(v)$. Consequently,Thus $u$ is additive. It follows successively that

An isometry is continuous. For any $\lambda \in \mathbb{R}$, choose rationals $q_j\to \lambda$. ThenTogether with additivity, this proves that $u$ is real-linear. Hence the origin-fixing case of the Mazur-Ulam theorem givesfor all $a,b\in \mathbb{R}$ and $v,w\in V$.