Closed-graph theorem

Let $X$ and $Y$ be complete metric linear spaces with translation-invariant metrics, i.e. $\rho_X(x_1+a,x_2+a) = \rho_X(x_1,x_2)$, $x_1,x_2,a \in X$ (similarly for $Y$), and let $T$ be a linear operator from $X$ to $Y$. If the graph $\mathrm{Gr}(T) = \{ (x,Tx) : x \in X \}$ of this operator is a closed subset of the Cartesian product $X \times Y$, then $T$ is continuous. The closed-graph theorem has various generalizations; for example: a linear mapping with closed graph from a separable barrelled space into a perfectly-complete space is continuous. Closely related theorems are the open-mapping theorem and Banach's homeomorphism theorem.

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Cf. also Open-mapping theorem (also for the Banach homeomorphism theorem).