Exercise 7, § 2

Let E be a Hilbert space having an orthonormal basis $(e_s)_{s \in \mathbf{R}}$ equipotent to $\mathbf{R}$. For every $s \in \mathbf{R}$, denote by $U(s)$ the isometry of E such that $U(s) \cdot e_t = e_{s+t}$ for all $t \in \mathbf{R}$; the linear representation $s \mapsto U(s)$ of $\mathbf{R}$ in E is not continuous, even though the set of $U(s)$ is equicontinuous.