Let $n\in\mathbb N$, and let $G\le GL(n,\mathbb C)$ be a subgroup that is closed as a subset of $M(n,\mathbb C)$ with its Euclidean topology. Define
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\begin{align*}
\mathfrak g:=\{X\in M(n,\mathbb C):\exp(tX)\in G\text{ for every }t\in\mathbb R\},
\end{align*}
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regarded as a real [vector space](/page/Vector%20Space). Then $G$ admits a smooth manifold structure compatible with its [subspace topology](/page/Subspace%20Topology), of real dimension $\dim_{\mathbb R}\mathfrak g$, such that the multiplication map
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\begin{align*}
G\times G&\to G
\end{align*}
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\begin{align*}
(a,b)&\mapsto ab
\end{align*}
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and the inversion map
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\begin{align*}
G&\to G
\end{align*}
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\begin{align*}
a&\mapsto a^{-1}
\end{align*}
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are smooth.
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More precisely, for every sufficiently small open neighbourhood $U\subset\mathfrak g$ of $0$ such that $\exp|_U$ is injective and $V:=\exp(U)$ is a neighbourhood of $I$ in $G$, the maps
for $a\in G$ are smooth coordinate charts and form a smooth atlas. With this smooth structure, the tangent space $T_I G$ is naturally identified with $\mathfrak g$.