course |
course_year |
question_number |
tags |
title |
year |
Analysis II |
IB |
0 |
|
Paper 2, Section I, G |
2010 |
Let $c>1$ be a real number, and let $F_{c}$ be the space of sequences $\mathbf{a}=\left(a_{1}, a_{2}, \ldots\right)$ of real numbers $a_{i}$ with $\sum_{r=1}^{\infty} c^{-r}\left|a_{r}\right|$ convergent. Show that $|\mathbf{a}|{c}=\sum{r=1}^{\infty} c^{-r}\left|a_{r}\right|$ defines a norm on $F_{c}$.
Let $F$ denote the space of sequences a with $\left|a_{i}\right|$ bounded; show that $F \subset F_{c}$. If $c^{\prime}>c$, show that the norms on $F$ given by restricting to $F$ the norms $|\cdot|{c}$ on $F{c}$ and $|\cdot|{c^{\prime}}$ on $F{c^{\prime}}$ are not Lipschitz equivalent.
By considering sequences of the form $\mathbf{a}^{(n)}=\left(a, a^{2}, \ldots, a^{n}, 0,0, \ldots\right)$ in $F$, for $a$ an appropriate real number, or otherwise, show that $F$ (equipped with the norm $|.|_{c}$ ) is not complete.