(i) Let $p$ be a point of the compact interval $I=[a, b] \subset \mathbb{R}$ and let $\delta_{p}: C(I) \rightarrow \mathbb{R}$ be defined by $\delta_{p}(f)=f(p)$. Show that

$$\delta_{p}:\left(C(I),\|\cdot\|_{\infty}\right) \rightarrow \mathbb{R}$$

is a continuous, linear map but that

$$\delta_{p}:\left(C(I),\|\cdot\|_{1}\right) \rightarrow \mathbb{R}$$

is not continuous.

(ii) Consider the space $C^{(n)}(I)$ of $n$-times continuously differentiable functions on the interval $I$. Write

$$\|f\|_{\infty}^{(n)}=\sum_{k=0}^{n}\left\|f^{(k)}\right\|_{\infty} \quad \text { and } \quad\|f\|_{1}^{(n)}=\sum_{r=0}^{n}\left\|f^{(k)}\right\|_{1}$$

for $f \in C^{(n)}(I)$. Show that $\left(C^{(n)}(I),\left\|^{\prime} \cdot\right\|_{\infty}^{(n)}\right)$ is a complete normed space. Is the space $\left(C^{(n)}(I),\|\cdot\|_{1}^{(n)}\right)$ also complete?

Let $f: I \rightarrow I$ be an $n$-times continuously differentiable map and define

$$\mu_{f}: C^{(n)}(I) \rightarrow C^{(n)}(I) \quad \text { by } \quad g \mapsto g \circ f .$$

Show that $\mu_{f}$ is a continuous linear map when $C^{(n)}(I)$ is equipped with the norm $\|\cdot\|_{\infty}^{(n)}$.