Proposition 5

Let $G_1$ be a topological group, let $G_2$ be a complete Hausdorff topological group, and let $H_1$ (resp. $H_2$) be a dense subgroup of $G_1$ (resp. $G_2$). Then every continuous homomorphism $u$ of $H_1$ into $H_2$ can be uniquely extended to a continuous homomorphism $\bar{u}$ of $G_1$ into $G_2$. Furthermore, if $G_1$ is Hausdorff and complete, and if $u$ is an isomorphism of $H_1$ onto $H_2$, then $\bar{u}$ is an isomorphism of $G_1$ onto $G_2$.

$\bar{u}$ is uniformly continuous with respect to the right uniformities of $H_1$ and $H_2$ (no. 1, Proposition 3), hence admits a unique extension to a mapping $\bar{u}$ of $G_1$ into $G_2$ which is uniformly continuous with respect to the right uniformities of these groups (Chapter II, § 3, no. 6, Theorem 2). Moreover, by virtue of the principle of extension of identities (Chapter I, § 8, no. 1, Corollary 1 to Proposition 2), $\bar{u}$ is a homomorphism of $G_1$ into $G_2$, whence the first assertion of the Proposition. To prove the second assertion, it is enough to consider the isomorphism $v$ of $H_2$ onto $H_1$, which is the inverse of $u$, and its extension $\bar{v}$ to a continuous homomorphism of $G_2$ into $G_1$; by reason of the uniqueness of the extension, $\bar{v} \circ \bar{u}$ and $\bar{u} \circ \bar{v}$ are the identity mappings of $G_1$ and $G_2$ respectively, and therefore (Set Theory, R, § 2, no. 12) $\bar{u}$ is bijective.