Let $X$ be a [Banach space](/page/Banach%20Space), let $A: D(A) \subset X \to X$ be a densely defined linear operator, and suppose that $A$ generates a strongly continuous semigroup $(T(t))_{t \ge 0}$ on $X$, with $T(t) \in \mathcal{L}(X)$ for every $t \ge 0$. Then, for every $\tau > 0$, there exists a constant $M_\tau > 0$ such that, for all $u_0, v_0 \in X$ and all $t \in [0,\tau]$,

\begin{align*} \|T(t)u_0 - T(t)v_0\|_X \le M_\tau \|u_0 - v_0\|_X. \end{align*}