Normalized velocity-reset projection
= Normalized velocity-reset projection
{title2=$Pg=\rho(g)M,\quad P^2=P$}
For a nonnegative normalized velocity density $M$, $Pg(x,v)=M(v)\int g(x,w)dw$ is a positive <norm>-one projection on phase-space $L^1$. Normalization gives $P^2=P$. Its range consists of fields $a(x)M(v)$. This Banach-space projection need not be an <orthogonal projection> in unweighted $L^2$; no such assertion is needed for the $L^1$ collision estimates.