| course |
course_year |
question_number |
tags |
title |
year |
Analysis II |
IB |
4 |
|
Paper 2, Section II, G |
2017 |
Let $V$ be a real vector space. What is a norm on $V$ ? Show that if $|-|$ is a norm on $V$, then the maps $T_{v}: x \mapsto x+v\left(\right.$ for $v \in V$ ) and $m_{a}: x \mapsto a x$ (for $a \in \mathbb{R}$ ) are continuous with respect to the norm.
Let $B \subset V$ be a subset containing 0 . Show that there exists at most one norm on $V$ for which $B$ is the open unit ball.
Suppose that $B$ satisfies the following two properties:
-
if $v \in V$ is a nonzero vector, then the line $\mathbb{R} v \subset V$ meets $B$ in a set of the form ${t v:-\lambda<t<\lambda}$ for some $\lambda>0$;
-
if $x, y \in B$ and $s, t>0$ then $(s+t)^{-1}(s x+t y) \in B$.
Show that there exists a norm $|-|_{B}$ for which $B$ is the open unit ball.
Identify $|-|_{B}$ in the following two cases:
(i) $V=\mathbb{R}^{n}, B=\left{\left(x_{1}, \ldots, x_{n}\right) \in \mathbb{R}^{n}:-1<x_{i}<1\right.$ for all $\left.i\right}$.
(ii) $V=\mathbb{R}^{2}, B$ the interior of the square with vertices $(\pm 1,0),(0, \pm 1)$.
Let $C \subset \mathbb{R}^{2}$ be the set
$$C=\left{\left(x_{1}, x_{2}\right) \in \mathbb{R}^{2}:\left|x_{1}\right|<1,\left|x_{2}\right|<1, \text { and }\left(\left|x_{1}\right|-1\right)^{2}+\left(\left|x_{2}\right|-1\right)^{2}>1\right}$$
Is there a norm on $\mathbb{R}^{2}$ for which $C$ is the open unit ball? Justify your answer.