2014-3.md

course	course_year	question_number	tags	title	year
Analysis II	IB	3	IB 2014 Analysis II	Paper 1, Section II, F	2014

Define what it means for two norms on a real vector space $V$ to be Lipschitz equivalent. Show that if two norms on $V$ are Lipschitz equivalent and $F \subset V$, then $F$ is closed in one norm if and only if $F$ is closed in the other norm.

Show that if $V$ is finite-dimensional, then any two norms on $V$ are Lipschitz equivalent.

Show that $|f|{1}=\int{0}^{1}|f(x)| d x$ is a norm on the space $C[0,1]$ of continuous realvalued functions on $[0,1]$. Is the set $S={f \in C[0,1]: f(1 / 2)=0}$ closed in the norm $|\cdot| 1$ ?

Determine whether or not the norm $|\cdot|{1}$ is Lipschitz equivalent to the uniform $\operatorname{norm}|\cdot|{\infty}$ on $C[0,1]$.

[You may assume the Bolzano-Weierstrass theorem for sequences in $\mathbb{R}^{n}$.]