The quotient-manifold characterization of Banach–Lie subgroups

Let $G$ be a Banach–Lie group and let $H$ be a closed subgroup. The quotient set $G/H$ is formed from the left cosets $gH$, and the quotient map is

$$q \: G \to G/H,\qquad g\mapsto gH.$$

A closed subgroup $H$ of a Banach–Lie group $G$ is a Banach–Lie subgroup when it is a Banach–Lie group with respect to the subspace topology. Quotient-manifold characterization. The subgroup $H$ is a Banach–Lie subgroup if and only if $G/H$ carries the structure of a Banach manifold such that the quotient map $q$ has surjective differential at every point and the action of $G$ on $G/H$ is smooth. This would provide a criterion for recognizing general closed subgroups as Banach–Lie subgroups, extending the corresponding characterization known for closed normal subgroups.