Let $C[0,1]$ be the space of continuous real-valued functions on $[0,1]$, and let $d_{1}, d_{\infty}$ be the metrics on it given by

$$d_{1}(f, g)=\int_{0}^{1}|f(x)-g(x)| d x \quad \text { and } \quad d_{\infty}(f, g)=\max _{x \in[0,1]}|f(x)-g(x)|$$

Show that id : $\left(C[0,1], d_{\infty}\right) \rightarrow\left(C[0,1], d_{1}\right)$ is a continuous map. Do $d_{1}$ and $d_{\infty}$ induce the same topology on $C[0,1]$ ? Justify your answer.

Let $d$ denote for any $m \in \mathbb{N}$ the uniform metric on $\mathbb{R}^{m}: d\left(\left(x_{i}\right),\left(y_{i}\right)\right)=\max _{i}\left|x_{i}-y_{i}\right|$. Let $\mathcal{P}_{n} \subset C[0,1]$ be the subspace of real polynomials of degree at most $n$. Define a Lipschitz map between two metric spaces, and show that evaluation at a point gives a Lipschitz map $\left(C[0,1], d_{\infty}\right) \rightarrow(\mathbb{R}, d)$. Hence or otherwise find a bijection from $\left(\mathcal{P}_{n}, d_{\infty}\right)$ to $\left(\mathbb{R}^{n+1}, d\right)$ which is Lipschitz and has a Lipschitz inverse.

Let $\tilde{\mathcal{P}}_{n} \subset \mathcal{P}_{n}$ be the subset of polynomials with values in the range $[-1,1]$.

(i) Show that $\left(\tilde{\mathcal{P}}_{n}, d_{\infty}\right)$ is compact.

(ii) Show that $d_{1}$ and $d_{\infty}$ induce the same topology on $\tilde{\mathcal{P}}_{n}$.

Any theorems that you use should be clearly stated.

[You may use the fact that for distinct constants $a_{i}$, the following matrix is invertible:

$$\left(\begin{array}{ccccc} 1 & a_{0} & a_{0}^{2} & \ldots & a_{0}^{n} \\ 1 & a_{1} & a_{1}^{2} & \ldots & a_{1}^{n} \\ \vdots & \vdots & \vdots & & \vdots \\ 1 & a_{n} & a_{n}^{2} & \ldots & a_{n}^{n} \end{array}\right)$$