I've been mulling this idea over for a while now with little progress. Maybe someone here will have more success with it.
Here, let $\ell_1$ be the set of real-valued sequences $\{x_n\}$ such that $\sum_{n\in\mathbb{N}}|x_n|\lt\infty$, and let $\ell_\infty$ be the set of real-valued sequences such that $\sup_{n\in\mathbb{N}}|y_n|\lt\infty$. Suppose I equip $\ell_\infty$ with the coarsest topology such that all functions mapping $\{y_n\}\mapsto\sum_{n\in\mathbb{N}}x_ny_n$, for $\{x_n\}\in\ell_1$, are continuous from $\ell_\infty$ to $\mathbb{R}$.
Is there a proof that the set $\{\{y_n\}\mid \sup_{n\in\mathbb{N}}|y_n|\leq 1\}$ is compact? Thanks.
Edit: Thank you for the responses so far. I wanted to add one thing. Is it possible to show this in a way using only techniques of general topology? I haven't studied functional analysis, and would prefer not to resort to a theorem I haven't read about to convince myself of this. If not, I guess I know what I need to learn next.