Summary: In a barreled locally convex topological vector space \(E\) the pointwise convergence of a family \((T_\theta )_{\theta \in\mathbb R}\) of operators at a point \(\theta_0 \in\mathbb R\) implies the uniform convergence on compact subsets of \(E\) at \(\theta_0\). For differentiable one-parameter transformation groups in \(GL(E)\) we prove the stronger convergence property of regularity, namely, for each seminorm \(p\) of \(E\) there exists a seminorm \(q\) of \(E\) such that \[ \lim_{\theta\to 0}\sup_{q(x)\leq 1} p\left(\frac{T_\theta x - x}{\theta}- T^\prime x\right) = 0, \] where \(T^\prime\) is the infinitesimal generator of \((T_\theta )_{\theta \in \mathbb R}\). In particular the convergence is uniform on all bounded subsets of \(E\).