State the closed graph theorem.

(i) Let $X$ be a Banach space and $Y$ a vector space. Suppose that $Y$ is endowed with two norms $\|\cdot\|_{1}$ and $\|\cdot\|_{2}$ and that there is a constant $c>0$ such that $\|y\|_{2} \geqslant c\|y\|_{1}$ for all $y \in Y$. Suppose that $Y$ is a Banach space with respect to both norms. Suppose that $T: X \rightarrow Y$ is a linear operator, and that it is bounded when $Y$ is endowed with the $\|\cdot\|_{1}$ norm. Show that it is also bounded when $Y$ is endowed with the $\|\cdot\|_{2}$ norm.

(ii) Suppose that $X$ is a normed space and that $\left(x_{n}\right)_{n=1}^{\infty} \subseteq X$ is a sequence with $\sum_{n=1}^{\infty}\left|f\left(x_{n}\right)\right|<\infty$ for all $f$ in the dual space $X^{*}$. Show that there is an $M$ such that

$$\sum_{n=1}^{\infty}\left|f\left(x_{n}\right)\right| \leqslant M\|f\|$$

for all $f \in X^{*}$.

(iii) Suppose that $X$ is the space of bounded continuous functions $f: \mathbb{R} \rightarrow \mathbb{R}$ with the sup norm, and that $Y \subseteq X$ is the subspace of continuously differentiable functions with bounded derivative. Let $T: Y \rightarrow X$ be defined by $T f=f^{\prime}$. Show that the graph of $T$ is closed, but that $T$ is not bounded.