Are there two norms in any space that neither of them is subordinate to the other?

Question

Can you prove if no or give an example if yes?

Obviously, this should be an example of an infinite-dimensional space, since in a finite-dimensional space, any two norms are equivalent.

Answer 1

On the space $X$ of finitely non-zero sequences let $\|(x_n)\|_1=\sum a_n|x_n|$ and $\|(x_n)\|_2=\sum b_n|x_n|$ where $a_n>0$ and $b_n >0$ for all $n$. These are norms on $X$. Neither is sub-ordinate to the other if neither $\frac {a_n} {b_n}$ nor $\frac {b_n} {a_n}$ is bounded. I will let you construct such sequences.

Detailed solution: Take $a_n=1$ for $n$ even and $n$ for $n$ odd, $b_n=n$ for $n$ even and $1$ for $n$ odd. Prove by contradiction that inequalities like $\|.\|_1 \leq C \|.\|_2$ and $\|.\|_2 \leq C \|.\|_1$ cannot hold. If the first inequality holds put $(x_n)=e_n$ to see that $|a_n| \leq C|b_n|$ which is a contradiction. Second case is similar.