Let $0<p<r<1$. It's known that the dual space of $\ell_p$ is $\ell_\infty$. The problem is to study the relationship between the weak-* topologies $\sigma(\ell_\infty,\ell_p)$ and $\sigma(\ell_\infty,\ell_r)$. Also, when restricted to norm-bounded sets, they end up being equal. A hint is provided: use Alaoglu's theorem.
To begin with, let's try to find any relationship between the topologies. My first thought was to show any inclusion between the sequence spaces. It goes as it follows: if we take $x\in\ell_p$, then $\sum_{i=1}^\infty \vert x(i)\vert^p<\infty$. Then, from a certain point onwards, $\vert x(i)\vert^p<1$ and $\vert x(i)\vert<1$, thus $\vert x(i)\vert^p>\vert x(i)\vert^r$, and we conclude that $x\in\ell_r$. That way, we have more seminorms in $\sigma(\ell_\infty,\ell_r)$ so this one is bigger. I might need to show that the inclusion is strict. How do I do that? Any suggested sequence that their $r$ sum converges but the $p$ one doesn't?
Also, I don't know how would Alaoglu's theorem here. All we know is that the unit balls in either topologies are compact in the weak-* topologies. I've been tinkering with the identity map changing topologies that should be continuous if one is finer than the other, but I haven't deduced anything about the norm-bounded sets thing...
