Weakly null sine sequence in L1
= Weakly null sine sequence in L1
{c}
{title2=$\sin(nt)\rightharpoonup0$ in $L^1([0,2\pi])$}
The functions $f_n(t)=\sin(nt)$ converge weakly to zero in $L^1([0,2\pi])$. Indeed, every $g\in L^\infty$ also belongs to $L^1$ on this finite interval, and the <Riemann-Lebesgue lemma> gives $\int_0^{2\pi}g(t)\sin(nt)\,dt\to0$. However,
$$
\lVert f_n\rVert_1=\int_0^{2\pi}|\sin(nt)|\,dt=4,
$$
so $L^1([0,2\pi])$ does not have the <Schur property>.