Proving that the functional $T_x : (C[0,1], \| \cdot \|_1) \rightarrow \mathbb{R}$ is not bounded, where $T_x(f) = f(x)$ and $x \in [0,1]$.
$\delta : C([0,1])\to\mathbb{R}$, $f\mapsto f(0)$, with sup-norm is bounded but unbounded with one-norm.