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Summary: We prove a characterization of the immersions in the context of infinite dimensional manifolds with corners, we prove that a Hausdorff paracompact $C^p$-manifold whose charts are modelled over real Banach spaces which fulfil the Urysohn $C^p$-condition can be embedded in a real Banach space, $E$, by means of a closed embedding, $f$, such that, locally, its image is a totally neat submanifold of a quadrant of a closed vector subspace of $E$ and finally we prove that a Hausdorff paracompact topological space, $X$, is a Hilbert $C^\infty$-manifold without boundary if and only if $X$ is homeomorphic to $A$, where $A$ is a $C^\infty$-retract of an open set of a real Hilbert space.
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