  For suitable finite-dimensional smooth manifolds M (possibly with various
kinds of boundary or corners), locally convex topological vector spaces F and
non-negative integers k, we construct continuous linear operators S_n from the
space of F-valued k times continuously differentiable functions on M to the
corresponding space of smooth functions such that S_n(f) converges to f in
C^k(M,F) as n tends to infinity, uniformly for f in compact subsets of
C^k(M,F). We also study the existence of continuous linear right inverses for
restriction maps from C^k(M,F) to C^k(L,F) if L is a closed subset of M,
endowed with a C^k-manifold structure turning the inclusion map from L to M
into a C^k-map. Moreover, we construct continuous linear right inverses for
restriction operators between spaces of sections in vector bundles in many
situations, and smooth local right inverses for restriction operators between
manifolds of mappings. We also obtain smoothing results for sections in fibre
bundles.
