Hey I'm really stuck on what I think is an interesting 'paradox'. Consider the sequence of functions $f_n = 1_{[n,n+1]}$ (indicator functions of the interval $[n,n+1]$.
These are uniformly bounded in the $L^1$ norm. It follows that, considering $L^1 \subset (L^1)^{**}$, that this belongs to a weak-*
compact set (by the banach alaoglu theorem). This should mean that there is a weak-*
convergent NET. You can see easily there is no weak-*
convergent subsequence: consider just $g=1 \in L^{\infty}$ then $\int f_n g = 1$ always.
My question is, what is going on here? Compactness of the weak-* unit ball is still true, but what does it mean in this case? Does it mean that every neighborhood of 0 in the weak-* topology intersects some of my functions f_n?