Let $G \le GL(n,\mathbb C)$ be a matrix Lie group, where this means that $G$ is a closed subgroup of $GL(n,\mathbb C)$ in the [subspace topology](/page/Subspace%20Topology) inherited from $M(n,\mathbb C)$. Then $G$ admits a smooth structure for which $G$ is an embedded submanifold of the open manifold $GL(n,\mathbb C)$. Consequently, through the open inclusion $GL(n,\mathbb C)\subset M(n,\mathbb C)$, the group $G$ is also an embedded submanifold of the real [vector space](/page/Vector%20Space) $M(n,\mathbb C)$. With this smooth structure, the multiplication map

\begin{align*} m_G:G\times G\to G,\quad (A,B)\mapsto AB \end{align*}

and the inversion map

\begin{align*} i_G:G\to G,\quad A\mapsto A^{-1} \end{align*}

are smooth.