An invariant pseudometric \(\rho\) on an abelian group \(G\) is said to have the Enflo property provided \(\forall_{x\in G}[\rho(x^2,e)=2\rho(x,e)].\) It is proved that a Hausdorff abelian group can be embedded as a topological subgroup in a locally convex Hausdorff topological vector space \(\Leftrightarrow\) the topology of \(G\) is generated by a family of pseudometrics with the Enflo property. If \(X\) is a Tikhonov space, \(e\in X\) and \(F\) is the free abelian group generated by \(X\setminus\{e\},\rho\) a continuous pseudometric on \(X,\) then the Graev extension \(\rho^\prime\) of \(\rho\) on \(F\) has the Enflo property. Main result (Theorem 2.6): Let \(X\) be a Tikhonov space and let \(\text{FLCS} (X)\) be the free locally convex topological vector space on \(X.\) Then the subgroup of \(\text{FLCS} (X)\) that is algebraically generated by \(X\) with its induced topology is topologically isomorphic to the free abelian topological group on \(X.\)