In this paper we prove the existence of a closed neat embedding of a Hausdorff paracompact Hilbert manifold with smooth boundary into $H\times [0,+\infty )$, where $H$ is a Hilbert space, such that the normal space in each point of a certain neighbourhood of the boundary is contained in $H\times \left\{0\right\}$. Then, we give a neccesary and sufficient condition that a Hausdorff paracompact topological space could admit a differentiable structure of class $\infty$ with smooth boundary.

Let M be a separable $C^{\infty }$ Finsler manifold of infinite dimension. Then it is proved, amongst other results, that under suitable conditions of local extensibility the germ of a $C^{\infty }$ function, or of a $C^{\infty }$ section of a vector bundle, on the union of a closed submanifold and a closed locally compact set in M, extends to a $C^{\infty }$ function on the whole of M.