Abelian extensions of topological vector groups

The possibility of endowing an Abelian topological group G with the structure of a topological vector space when a subgroup F of G and the quotient group G F are topological vector groups is investigated. It is shown that, if F is a real Fréchet group and G F a complete metrizable real vector group, then G is a complete metrizable real vector group. This result is of particular interest if G F is finite dimensional or if F is one dimensional and G F a separable Hilbert group.