Past exam of the analysis course of the University of Oxford / 2011 / 4 / a

The graph $G(T)=\{(x,Tx):x\in X\}$ is closed if it is a closed subset of $X\times Y$. Equivalently, $x_n\to x$ and $T{x}_n\to y$ imply $Tx=y$.

The closed graph theorem states that an everywhere-defined linear map between Banach spaces is bounded if and only if its graph is closed. Completeness of both spaces is part of the theorem. A bounded map has a closed graph by continuity.

Conversely, if the graph is closed, it is a Banach space in the product norm $\Vert (x,y)\Vert =\Vert x\Vert +\Vert y\Vert$. The coordinate projection $P:G(T)\to X$, $P(x,Tx)=x$, is a bounded linear bijection. The open mapping theorem makes $P^{-1}$ bounded. The other coordinate projection $Q:G(T)\to Y$ is bounded, and $T=Q{P}^{-1}$ is therefore bounded. This proves the theorem from the permitted result.