2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/160048In this note we show that for any proper action of a Banach--Lie group $G$ on a Banach manifold $M$, the corresponding tangent maps $\g \to T_x(M)$ have closed range for each $x \in M$, i.e., the tangent spaces of the orbits are closed. As a consequence, for each free proper action on a Hilbert manifold, the quotient $M/G$ carries a natural manifold structure.Differential GeometryDynamical Systems22E65; 58B25; 57E20text