**Proof plan.** The proof reduces to the [open mapping theorem for Fréchet spaces](/theorems/712). The graph $G(T)$ is a closed subspace of the Fréchet space $X \times Y$, hence itself a Fréchet space. The projection $\pi_X: G(T) \to X$ is a continuous linear bijection between [Fréchet spaces](/page/Fr%C3%A9chet%20Space), so the [open mapping theorem](/theorems/631) makes $\pi_X$ an isomorphism. The [continuity](/page/Continuity) of $T$ follows from writing $T = \pi_Y \circ \pi_X^{-1}$. **Step 1 (The product $X \times Y$ is Fréchet).** Let $\{p_n\}_{n=1}^\infty$ and $\{q_n\}_{n=1}^\infty$ be the generating seminorm families for $X$ and $Y$ respectively. The product topology on $X \times Y$ is generated by the countable separating family $\{r_n\}_{n=1}^\infty$ where $r_n(x, y) = p_n(x) + q_n(y)$ (or equivalently, $\max(p_n(x), q_n(y))$). Completeness of $X \times Y$ follows from the characterisation in the [metrizability theorem](/theorems/708): a [sequence](/page/Sequence) $\{(x_k, y_k)\}$ is Cauchy in every $r_n$ if and only if $\{x_k\}$ is Cauchy in every $p_n$ and $\{y_k\}$ is Cauchy in every $q_n$, so each component converges by completeness of $X$ and $Y$. **Step 2 ($G(T)$ is a Fréchet space).** The graph $G(T) = \{(x, Tx) : x \in X\}$ is a closed subspace of $X \times Y$ by hypothesis. A closed subspace of a complete metrizable locally convex space is complete and metrizable with the induced topology (a [Cauchy sequence](/page/Cauchy%20Sequence) in the subspace is Cauchy in the ambient space, hence convergent, and the [limit](/page/Limit) lies in the subspace because it is closed). Therefore $G(T)$ is a Fréchet space. **Step 3 (The projection $\pi_X$ is a continuous bijection).** [claim:The Projection Is A Continuous Bijection] The first projection $\pi_X: G(T) \to X$ defined by $\pi_X(x, Tx) = x$ is a continuous linear bijection. [/claim] [proof] Linearity is immediate. Injectivity: if $\pi_X(x, Tx) = \pi_X(x', Tx') = 0$, then $x = x' = 0$ and $Tx = Tx' = 0$. Surjectivity: for every $x \in X$, the pair $(x, Tx) \in G(T)$ and $\pi_X(x, Tx) = x$. Continuity: for every generating seminorm $p_n$ on $X$, we have $p_n(\pi_X(x, Tx)) = p_n(x) \le r_n(x, Tx)$, so $\pi_X$ is continuous. [/proof] **Step 4 (Applying the open mapping theorem).** By the [open mapping theorem for Fréchet spaces](/theorems/712), the continuous surjective linear map $\pi_X: G(T) \to X$ between Fréchet spaces is open. A continuous, open, linear bijection is a [topological](/page/Topology) isomorphism, so $\pi_X^{-1}: X \to G(T)$ is continuous. **Step 5 (Conclusion).** The second projection $\pi_Y: G(T) \to Y$, given by $\pi_Y(x, Tx) = Tx$, is continuous (by the same argument as for $\pi_X$). Therefore $T = \pi_Y \circ \pi_X^{-1}$ is continuous as a composition of continuous [linear maps](/page/Linear%20Map). $\blacksquare$