Nemytskii operator
= Nemytskii operator
{c}
{title2=$N_f(u)(x)=f(u(x))$}
{wiki}
= Superposition operator
{synonym}
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 $\|N_f(u)\|_2\leq\|f\|_\infty|\Omega|^{1/2}$. 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 $L^2$.