Subspaces of the free topological vector space on the unit interval

For a Tychonoff space X, let V(X) be the free topological vector space over X, A(X) the free abelian topological group over X and I the unit interval with its usual topology. It is proved here that if X is a subspace of I, then the following are equivalent: V(X) can be embedded in V(I) as a topological vector subspace; A(X) can be embedded in A(I) as a topological subgroup; X is locally compact.