Closed Graph Theorem

Theorem
Let $\struct {X, \norm {\,\cdot\,}_X}$ and $\struct {Y, \norm {\,\cdot\,}_Y}$ be Banach spaces.

Let $T : X \to Y$ be a linear transformation.

Let $G_T \subseteq X \times Y$ be the graph of $T$.

Suppose that:


 * $G_T$ is closed in the direct product $X \times Y$ equipped with the direct product norm $\norm \cdot_{X \times Y}$.

Then:


 * $T$ is bounded.