Closedness of the Tangent Spaces to the Orbits of Proper Actions

In 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 -> T-x(M) have closed range for each x is an element of 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.