2013-5.md

course	course_year	question_number	tags	title	year
Analysis II	IB	5	IB 2013 Analysis II	Paper 3, Section II, F	2013

For each of the following statements, provide a proof or justify a counterexample.

1. The norms $|x|{1}=\sum{i=1}^{n}\left|x_{i}\right|$ and $|x|_{\infty}=\max {1 \leqslant i \leqslant n}\left|x{i}\right|$ on $\mathbb{R}^{n}$ are Lipschitz equivalent.

2. The norms $|x|{1}=\sum{i=1}^{\infty}\left|x_{i}\right|$ and $|x|{\infty}=\max {i}\left|x{i}\right|$ on the vector space of sequences $\left(x{i}\right){i \geqslant 1}$ with $\sum\left|x{i}\right|<\infty$ are Lipschitz equivalent.

3. Given a linear function $\phi: V \rightarrow W$ between normed real vector spaces, there is some $N$ for which $|\phi(x)| \leqslant N$ for every $x \in V$ with $|x| \leqslant 1$.

4. Given a linear function $\phi: V \rightarrow W$ between normed real vector spaces for which there is some $N$ for which $|\phi(x)| \leqslant N$ for every $x \in V$ with $|x| \leqslant 1$, then $\phi$ is continuous.

5. The uniform norm $|f|=\sup _{x \in \mathbb{R}}|f(x)|$ is complete on the vector space of continuous real-valued functions $f$ on $\mathbb{R}$ for which $f(x)=0$ for $|x|$ sufficiently large.

6. The uniform norm $|f|=\sup _{x \in \mathbb{R}}|f(x)|$ is complete on the vector space of continuous real-valued functions $f$ on $\mathbb{R}$ which are bounded.