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In a barreled locally convex topological vector space E the pointwise convergence of a family (Tθ)θ∈R of operators at a point θ0 ∈ R implies the uniform convergence on compact subsets of E at θ0. 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 θ→0 sup q(x)≤1 p ( Tθx− x θ − T ′x ) = 0, where T ′ is the infinitesimal generator of (Tθ)θ∈R. In particular the convergence is uniform on all bounded subsets of E.

On the regularity of one parameter transformation groups in barreled locally convex topological vector spaces