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?

separablespaces, so that their closed unit ball is weak* compact and metrizable. Indeed, $f_n$ is weak* convergent to zero for all $p > 1$. – Pietro Majer Jul 19 '10 at 6:35