Confusion about bounded nets and weak-* convergence vs. subsequences in reflexive spaces

Question

I am confused about something. I know that the following general fact always holds: Bounded nets and weak- convergence in the bidual of a Banach space*.enter link description here

"Every bounded net in a Banach space $Y$ has a subnet that converges in $Y^{∗∗}$ with respect to the weak-∗ topology."

On the other hand, in the book Análise Funcional by Geraldo Botelho, Theorem 6.5.4 states that in a reflexive space, every bounded sequence has a weakly convergent subsequence.

My question is: sequences are a particular case of nets, so shouldn't this theorem be a particular case of the general fact? Moreover, I understand that reflexivity is crucial when dealing with sequences instead of nets, but I can't quite see how this difference plays a role here.

Is there something I am missing? Could someone clarify this for me?

I would really appreciate any help!

Answer 1

The subtle point here is that, while every sequence is a net, not every subnet of a sequence is a subsequence. An example of such a subnet given in this answer.

For this reason, in general compactness does not imply sequential compactness. As it happens, for the weak topology on Banach spaces they are equivalent, though you don't need such heavy machinery to prove the Theorem 6.5.4 you link (as the proof demonstrates).