| course |
course_year |
question_number |
tags |
title |
year |
Analysis II |
IB |
4 |
|
Paper 4, Section II, G |
2016 |
(a) Let $V$ be a real vector space. What does it mean to say that two norms on $V$ are Lipschitz equivalent? Prove that every norm on $\mathbb{R}^{n}$ is Lipschitz equivalent to the Euclidean norm. Hence or otherwise, show that any linear map from $\mathbb{R}^{n}$ to $\mathbb{R}^{m}$ is continuous.
(b) Let $f: U \rightarrow V$ be a linear map between normed real vector spaces. We say that $f$ is bounded if there exists a constant $C$ such that for all $u \in U,|f(u)| \leqslant C|u|$. Show that $f$ is bounded if and only if $f$ is continuous.
(c) Let $\ell^{2}$ denote the space of sequences $\left(x_{n}\right){n \geqslant 1}$ of real numbers such that $\sum{n \geqslant 1} x_{n}^{2}$ is convergent, with the norm $\left|\left(x_{n}\right){n}\right|=\left(\sum{n \geqslant 1} x_{n}^{2}\right)^{1 / 2}$. Let $e_{m} \in \ell^{2}$ be the sequence $e_{m}=\left(x_{n}\right){n}$ with $x{m}=1$ and $x_{n}=0$ if $n \neq m$. Let $w$ be the sequence $\left(2^{-n}\right){n}$. Show that the subset ${w} \cup\left{e{m} \mid m \geqslant 1\right}$ is linearly independent. Let $V \subset \ell^{2}$ be the subspace it spans, and consider the linear map $f: V \rightarrow \mathbb{R}$ defined by
$$f(w)=1, \quad f\left(e_{m}\right)=0 \quad \text { for all } m \geqslant 1 .$$
Is $f$ continuous? Justify your answer.