2015-1.md

course	course_year	question_number	tags	title	year
Analysis II	IB	1	IB 2015 Analysis II	Paper 4, Section I, G	2015

Define what is meant for two norms on a vector space to be Lipschitz equivalent.

Let $C_{c}^{1}([-1,1])$ denote the vector space of continuous functions $f:[-1,1] \rightarrow \mathbb{R}$ with continuous first derivatives and such that $f(x)=0$ for $x$ in some neighbourhood of the end-points $-1$ and 1 . Which of the following four functions $C_{c}^{1}([-1,1]) \rightarrow \mathbb{R}$ define norms on $C_{c}^{1}([-1,1])$ (give a brief explanation)?

Among those that define norms, which pairs are Lipschitz equivalent? Justify your answer.