[proofplan]
We first recall that the ambient general linear groups carry their standard real Lie group structures as open subsets of finite-dimensional matrix spaces. For a closed subgroup of $GL(n,\mathbb C)$, [Cartan's closed subgroup theorem](/theorems/8813) applies directly and gives an embedded matrix Lie subgroup structure. For a closed subgroup of $GL(n,\mathbb R)$, we either apply the same real version of Cartan's theorem inside $GL(n,\mathbb R)$, or equivalently regard the subgroup as closed inside $GL(n,\mathbb C)$ while keeping the real matrix structure. The conclusion is that the subgroup inherits precisely the embedded real Lie group structure required in the definition of a matrix Lie group.
[/proofplan]
[step:Place the ambient general linear groups in their standard Lie group structures]
By [citetheorem:8769], for $\mathbb F\in\{\mathbb R,\mathbb C\}$ the set $GL(n,\mathbb F)$ is an open subset of the finite-dimensional [vector space](/page/Vector%20Space) $M(n,\mathbb F)$ and is a Lie group under matrix multiplication.
When $\mathbb F=\mathbb C$, we regard $M(n,\mathbb C)$ as a real vector space of dimension $2n^2$, so $GL(n,\mathbb C)$ is a real Lie group. When $\mathbb F=\mathbb R$, the space $M(n,\mathbb R)$ is a real vector space of dimension $n^2$, so $GL(n,\mathbb R)$ is a real Lie group. Thus both ambient groups satisfy the Lie group hypothesis needed for Cartan's closed subgroup theorem.
[/step]
[step:Apply Cartan's closed subgroup theorem in the complex ambient group]
Let $H\le GL(n,\mathbb C)$ be a subgroup that is closed in the [subspace topology](/page/Subspace%20Topology) inherited from $M(n,\mathbb C)$. Since $GL(n,\mathbb C)$ is an open subset of $M(n,\mathbb C)$, the topology inherited by $H$ from $M(n,\mathbb C)$ agrees with the topology inherited by $H$ from the Lie group $GL(n,\mathbb C)$.
The hypotheses of [citetheorem:8813] are therefore satisfied with $G=GL(n,\mathbb C)$ and with this subgroup $H$: the ambient group is a matrix Lie group, and $H$ is closed as a subgroup of it. Hence $H$ is an embedded matrix Lie subgroup of $GL(n,\mathbb C)$. In particular, $H$ is a matrix Lie group, regarded as a real Lie group.
[guided]
Let us spell out exactly where each hypothesis is used. The subgroup under consideration is a map-level inclusion
\begin{align*}
\iota_H:H\to GL(n,\mathbb C),
\end{align*}
where $\iota_H(A)=A$ for every $A\in H$. The theorem assumes that $H$ is closed in the subspace topology inherited from $M(n,\mathbb C)$. Since $GL(n,\mathbb C)$ is open in $M(n,\mathbb C)$ by [citetheorem:8769], the subspace topology on $H$ coming from $M(n,\mathbb C)$ is the same as the subspace topology on $H$ coming from $GL(n,\mathbb C)$.
Now we apply [citetheorem:8813]. Its hypotheses require a matrix Lie group $G$ and a closed subgroup of $G$. We take
\begin{align*}
G=GL(n,\mathbb C).
\end{align*}
The first step verified that $G$ is a real matrix Lie group. The subgroup hypothesis is exactly $H\le GL(n,\mathbb C)$, and the preceding paragraph verifies closedness in the topology of $G$. Cartan's theorem therefore gives $H$ an embedded Lie subgroup structure inside $GL(n,\mathbb C)$.
This is the correct conclusion for the complex case: it makes $H$ a real matrix Lie group embedded in the complex matrix group. We are not asserting that $H$ is necessarily a complex Lie subgroup, since a closed subgroup of $GL(n,\mathbb C)$ need not carry a compatible complex manifold structure.
[/guided]
[/step]
[step:Apply the real version to closed subgroups of $GL(n,\mathbb R)$]
Let $K\le GL(n,\mathbb R)$ be closed in the subspace topology inherited from $M(n,\mathbb R)$. By the real case of [citetheorem:8769], the ambient group $GL(n,\mathbb R)$ is a real matrix Lie group. Since $GL(n,\mathbb R)$ is open in $M(n,\mathbb R)$, the topology inherited by $K$ from $M(n,\mathbb R)$ agrees with the topology inherited by $K$ from $GL(n,\mathbb R)$.
Applying [citetheorem:8813] in the real matrix group $G=GL(n,\mathbb R)$, the subgroup $K$ obtains an embedded real Lie subgroup structure. Thus $K$ is a real matrix Lie group embedded in $GL(n,\mathbb R)$.
[/step]
[step:Conclude both assertions]
The complex case shows that every closed subgroup of $GL(n,\mathbb C)$ is a real matrix Lie group embedded in the complex general linear group. The real case shows that every closed subgroup of $GL(n,\mathbb R)$ is a real matrix Lie group embedded in the real general linear group. These are exactly the two assertions of the theorem.
[/step]
