A pretty standard result says that if $T$ is a compact operator acting on a Banach space, then $T$ cannot be surjective. The proofs I've seen use the open mapping theorem and the fact that a topological vector space is locally compact if and only if it is finite dimensional. The open mapping theorem is usually phrased in terms of Banach spaces, although in can be phrased in terms of F-spaces and topological vector spaces. However, in both cases, completeness of the metric/norm seems essential. My question is,
Does there a compact operator on an infinite dimensional incomplete metric/normed vector space that is surjective?