= Concentrating spikes distinguish weak Lp and weak L1 compactness
{title2=$j^{1/p}\mathbf1_{(0,1/j)}\rightharpoonup0\ (p>1)$}
The displayed spikes have $L^p$ <norm> one. Pairing with $g\in L^{p'}$ and applying <Holder inequality> bounds the pairing by $(\int_0^{1/j}|g|^{p'})^{1/p'}$, which tends to zero. At exponent one, however, $j\mathbf1_{(0,1/j)}$ cannot converge weakly in $L^1$: every interval away from zero forces any limit to vanish there, while the constant <test function> preserves integral one. Their failure of <uniform integrability> records exactly the obstruction in the <Dunford-Pettis theorem>.
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