Let f be a smooth function defined on a finite union U of open convex sets in a locally convex Lindelöf space E. If, for every x ∈ U, the restriction of f to a suitable neighbourhood of x admits a smooth extension to the whole of E, then the restriction of f to a union of convex sets that is strictly smaller than U also admits a smooth extension to the whole of E.

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.