2019-6.md

course	course_year	question_number	tags	title	year
Analysis II	IB	6	IB 2019 Analysis II	Paper 2, Section II, 12E	2019

(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.