Failure of weak-star convergence to commute with squaring
= Failure of weak-star convergence to commute with squaring
In $L^\infty(\mathbb R)=(L^1(\mathbb R))^*$, the sequence $f_n(x)=\sin(nx)$ converges weak-star to zero by the <Riemann-Lebesgue lemma>, while $f_n^2$ converges weak-star to the constant $1/2$. Nonlinear pointwise operations therefore need not preserve weak-star limits.