= Concentration prevents weak compactness in L1
{title2=$f_j=j^n\mathbf1_{[0,1/j]^n}$}
The nonnegative <Lp space> functions $f_j$ have $L^1$ norm one. Any weak limit tested against indicators supported outside the origin must vanish almost everywhere. Testing against the constant bounded function one would nevertheless require integral one. This contradiction applies to every subnet whose indices tend to infinity, so the $L^1$ unit ball is not weakly compact. The abstract <Banach-Alaoglu theorem> remains valid: weak compactness of the $L^1$ unit ball is a different assertion from weak-star compactness of a dual ball.
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