Theorem 12.9.6 (Closed Graph Theorem).label Let $E, F$ be complete metric TVSs over $K \in \RC$ and $T \in \hom(E; F)$. If its graph $\Gamma(T) \subset E \times F$ is closed, then $T \in L(E; F)$.
Proof. Given that $E$ and $F$ are both complete metric TVSs, $E \times F$ is a complete metric TVS by Proposition 6.7.2. Since $\Gamma(T) \subset E \times F$ is a closed subspace of $E \times F$, it is also a complete metric TVS over $K$ by Proposition 6.7.3.
Let $\pi_{1}: E \times F \to E$ and $\pi_{2}: E \times F \to F$ be the projection maps of $E \times F$. As $\Gamma(T)$ is the graph of a function, $\pi_{1}|_{\Gamma(T)}: \Gamma(T) \to E$ is a continuous bijection. By the Open Mapping Theorem, it is an isomorphism. Therefore $T$ may be expressed as the following composition of continuous linear maps
$\square$
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