= Solution
Use the <weighted Hilbert structure of velocity-reset relaxation>, with <inner product> $\langle f,g\rangle=\int f\overline g/M$ and <norm> $\|f\|_H$. The <Cauchy-Schwarz inequality> gives
$$
\int|f|=\int\frac{|f|}{\sqrt M}\sqrt M\leq\|f\|_H\left(\int M\right)^{1/2}=\|f\|_H.
$$
Thus $\rho(f)$ is an absolutely convergent <integral> and a bounded <linear functional>. Since $\|M\|_H^2=\int M=1$, the <normalized velocity-reset collision operator> obeys
$$
\boxed{\|Lf\|_H=\|\rho(f)M-f\|_H\leq2\|f\|_H}.
$$
This establishes boundedness. The subsequent orthogonal decomposition improves the operator <norm> to exactly one.
Back to article page
