= Solution
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 $Tx_n\to y$ imply $Tx=y$.
The <closed graph theorem> states that \b[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> $\|(x,y)\|=\|x\|+\|y\|$. 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 $T=QP^{-1}$ is therefore bounded. This proves the theorem from the permitted result.
Back to article page
