I was reading through general-topology posts and I couldn't quite understand this one. I tried asking directly on the thread, but I didn't get a response.
This is in reference to Professor Israel's answer on How can I show this set of sequences is compact?. I followed all the logic up until the last sentence where he claims:
That is, $\iota(B)$ is the intersection of the closed sets $K(a,x,b,y)=\{p:ap_x+bp_y\}$ for $x,y\in\ell_1$ and $a,b\in\mathbb{R}$, so it is closed.
I'm probably missing something obvious, but why is $\iota(B)$ the intersection of those closed sets? The way I'm interpreting the claim is that $\iota(B)$ is a subset of $\mathbb{R}^{\ell_1}$ while the $K$'s are just a set of linear maps (if I understood that notation correctly) so $\iota(B)$ can't possibly be the intersection of them, much less compared with them since one is a set of elements and the other is a set of maps.
What am I missing?