Can anyone give an example of a relatively compact set in $\ell_p$ which is not majorable?

Question

Is there any relatively compact set in $\ell_p$ which is not majorable? If yes, can anyone give me an example?

Answer 1

Call $x^k(n)=\begin{cases}n^{-1/p}&\text{if }n=k\\ 0&\text{if }n\ne k\end{cases}$. Then $H=\{x^k\,:\, k\ge 1\}$ is the support of a sequence that converges to $0$, therefore it's relatively compact. However, in order for a vector $y$ to dominate all the elements of $H$ you must have $\lvert y(n)\rvert\ge n^{-1/p}$ for all $n\ge1$, and there is no such vector in $\ell^p$.