= Solution
The <bounded inverse theorem> states that a bounded bijective linear map between Banach spaces has a bounded inverse.
The <closed graph theorem> states that if $X$ and $Y$ are Banach spaces and $T:X\to Y$ is linear, then $T$ is continuous if and only if its graph
$$
\Gamma(T)=\{(x,Tx):x\in X\}
$$
is closed in the Banach space $X\times Y$.
If $T$ is continuous and $(x_n,Tx_n)\to(x,y)$, then $Tx_n\to Tx$, so uniqueness of limits gives $y=Tx$. Hence the graph is closed.
Conversely, suppose $\Gamma(T)$ is closed. It is then a Banach space. The coordinate projection
$$
P_X:\Gamma(T)\longrightarrow X,
\qquad (x,Tx)\longmapsto x,
$$
is bounded and bijective. By the bounded inverse theorem, $P_X^{-1}:x\mapsto(x,Tx)$ is bounded. Composing it with the bounded second-coordinate projection gives
$$
T=P_Y\circ P_X^{-1},
$$
so $T$ is continuous.
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