(a) (i) Define what it means for two norms on a vector space to be Lipschitz equivalent.

(ii) Show that any two norms on a finite-dimensional vector space are Lipschitz equivalent.

(iii) Show that if two norms $\|\cdot\|,\|\cdot\|^{\prime}$ on a vector space $V$ are Lipschitz equivalent then the following holds: for any sequence $\left(v_{n}\right)$ in $V,\left(v_{n}\right)$ is Cauchy with respect to $\|\cdot\|$ if and only if it is Cauchy with respect to $\|\cdot\|^{\prime}$.

(b) Let $V$ be the vector space of real sequences $x=\left(x_{i}\right)$ such that $\sum\left|x_{i}\right|<\infty$. Let

$$\|x\|_{\infty}=\sup \left\{\left|x_{i}\right|: i \in \mathbb{N}\right\},$$

and for $1 \leqslant p<\infty$, let

$$\|x\|_{p}=\left(\sum\left|x_{i}\right|^{p}\right)^{1 / p}$$

You may assume that $\|\cdot\|_{\infty}$ and $\|\cdot\|_{p}$ are well-defined norms on $V$.

(i) Show that $\|\cdot\|_{p}$ is not Lipschitz equivalent to $\|\cdot\|_{\infty}$ for any $1 \leqslant p<\infty$.

(ii) Are there any $p, q$ with $1 \leqslant p<q<\infty$ such that $\|\cdot\|_{p}$ and $\|\cdot\|_{q}$ are Lipschitz equivalent? Justify your answer.