Counter-example for closed mapping theorem in normed spaces
Solution 1:
Take any infinite dimensional Banach space $(X,\|\cdot\|)$, a discontinuous linear functional (this needs the axiom of choice) $f:X\to \mathbb R$, and define $\|x\|_1=\|x\|+|f(x)|$. Then $Y=X$ endowed with $\|\cdot\|_1$ is a normed space, and the identical map $T:X\to Y$ is unbounded (since $f$ is unbounded) with closed graph (because the inverse is continuous).