I was reading Haim Brezis and came across a small remark. It makes me become unsure of my understanding of definition of nets in topology.
The remark:With $E=l^{\infty}$ , the unit ball $B_{E^*}$ is compact with respect to its weak-$*$ topology, while we can construct a sequence in $B_{E^*}$ that don't have a convergent subsequence .
This mark reminds me of a result in topology:
Theorem : A topological space X is compact iff every net has a convergent subnet (1)
So , I tried to deduce something like below:
"For any $(x_n)_{n\in \mathbb{N}} \in B_{E^*}$. We have that $B_{E^*}$ is compact with respect to its weak-$*$ topology. Hence, by theorem (1), we get that, there is a convergent subsequence $(x_{n_k})_{k \in \mathbb{N}}$. Which is wrong.So I was wrong in some parts"
Can you help me figure out what I did wrong? Thank you
