Volterra series for the linear Boltzmann equation ($f={\sum }_{n\ge 0}{\tau }^{n}F$)

The factorial bound for a Volterra iterate gives convergence in the space of bounded strongly measurable $L^2$-valued functions on $[0,T]$. The sum solves $(I-\tau )f=F$, where $F=U_a(t,0)f_0$, and satisfies ${\sup }_{t\le T}\Vert f(t){\Vert }_2\le e^{CT}\Vert f_0{\Vert }_2$. Applying the same iterate bound to a difference $g=\tau g$ proves uniqueness on every finite interval.