2009-2.md

course	course_year	question_number	tags	title	year
Analysis II	IB	2	IB 2009 Analysis II	Paper 3, Section I, E	2009

What is meant by a norm on $\mathbb{R}^{n}$ ? For $\mathbf{x} \in \mathbb{R}^{n}$ write

$$\begin{gathered} |\mathbf{x}|{1}=\left|x{1}\right|+\left|x_{2}\right|+\cdots+\left|x_{n}\right| \ |\mathbf{x}|{2}=\sqrt{\left|x{1}\right|^{2}+\left|x_{2}\right|^{2}+\cdots+\left|x_{n}\right|^{2}} \end{gathered}$$

Prove that $|\cdot|{1}$ and $|\cdot|{2}$ are norms. [You may assume the Cauchy-Schwarz inequality.]

Find the smallest constant $C_{n}$ such that $|x|{1} \leqslant C{n}|x|{2}$ for all $x \in \mathbb{R}^{n}$, and also the smallest constant $C{n}^{\prime}$ such that $|x|{2} \leqslant C{n}^{\prime}|x|_{1}$ for all $x \in \mathbb{R}^{n}$.