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

We show that every real finite dimensional Hausdorff (not necessarily paracompact, not necessarily second countable) CrCr-manifold can be CrCr-embedded into a weakly complete vector space, i.e. a locally convex space of the form RIRI for an uncountable index set I and determine the minimal cardinality of I for which such an embedding is possible.