Let
- $E$ be a normed $\mathbb R$-vector space
- $C_0(E)$ denote the space of continuous functions vanishing at infinity
- $(T(t))_{t\ge0}$ be a contractive nonnegative semigroup on $C_0(E)$ with $$T_tf(x)\xrightarrow{t\to0}f(x)\;\;\;\text{for all }x\in E\text{ and }f\in C_0(E)\tag1.$$
Why can we conclude from $(1)$ that $(T(t))_{t\ge0}$ is strongly continuous?