Nemytskii operator

The Nemytskii operator associated with a scalar function $f$ applies $f$ pointwise to a function. The domain and target function spaces must be specified; an arbitrary continuous $f$ need not preserve them. On a finite-measure domain, a bounded continuous $f:\mathbb{R}\to \mathbb{R}$ defines a continuous map $N_f:L^2\to L^2$ with $\Vert N_f(u){\Vert }_2\le \Vert f{\Vert }_{\infty }|\Omega {|}^{1/2}$. To prove continuity, extract an almost-everywhere convergent further subsequence from every $L^2$-convergent subsequence and apply the dominated convergence theorem to $|f(u_j)-f(u){|}^2$. This forces convergence of the entire image sequence in $L^2$.