Smooth embeddings of the Long Line and other non-paracompact manifolds into locally convex spaces

We show that every finite dimensional Hausdorff (not necessarily paracompact, not necessarily second countable) $C^r$-manifold can be embedded into a weakly complete vector space, i.e. a locally convex topological vector space of the form ${\mathbb{R}}^I$ for an uncountable index set $I$ and determine the minimal cardinality of $I$ for which such an embedding is possible.