Real Analysis, Folland Problem 5.3.30 The Baire Category Theorem 
Problem 5.3.30 - Let $\mathscr{Y} = C([0,1])$ and $\mathscr{X} = C^{1}([0,1])$, both equipped with the uniform norm.
a.) $\mathscr{X}$ is not complete.
b.) The map $(d/dx):\mathscr{X}\rightarrow \mathscr{Y}$ is closed but not bounded.

For a.) I believe since $\mathscr{X}$ is equipped with the uniform norm that is we have $$\lVert C^{1}([0,1])\rVert_{\infty} \neq 0$$ so $\mathscr{X}$ cannot be complete. Not sure if I am defining this correctly, or if that is necessarily true all the time for the interval $[0,1]$.
For b.) I am confused on the mapping here. So I am not sure how to approach the problem.
Any insight or suggestions is greatly appreciated.
 A: The map you are supposed to consider is
$$
          \frac{d}{dx} : \mathscr{X}\rightarrow\mathscr{Y}
$$
where $\mathscr{X}$ is $C^1[0,1]$ equipped with the sup norm, and $\mathscr{Y}$ is $C[0,1]$ equipped with the sup norm. To show that $\frac{d}{dx}$ is closed, suppose that $\{ f_n\}\subset\mathscr{X}$ converges to some $f\in\mathscr{X}$, and suppose $\{g_n=\frac{d}{dx}f_n\}\subset\mathscr{Y}$ converges to some $g\in\mathscr{Y}$; you must show that $g=f'$. This follows rather easily from the fact that
$$
                 f_n(t)=f_n(0)+\int_{0}^{t}g_n(t)dt.
$$
Because $\{ f_n \}$ converges uniformly to $f$ and $\{ g_n\}$ converges uniformly to $g$, then
$$
                 f(t) = f(0)+\int_{0}^{t}g(t)dt.
$$
Therefore, $\frac{d}{dx}f = g$. So the operator $\frac{d}{dx}$ between these spaces is closed. If $\mathscr{X}$ were a Banach space, then the closed graph theorem would imply that $\frac{d}{dx}$ is bounded, which it is not (see comment from Henry W to your post.) Therefore $\mathscr{X}$ cannot be complete.
