Exercise 8.C.7

Proof. By Exercise 4 in section 5B, we have

It is easy to check that if $v\in \mathrm{range}<!--\; FUNCTION\; APPLICATION\; -->P$ then $\mathit{Pv}=v$. Thus

$(1)$ and 8.26 then imply that these inclusions are actually equalities, which gives the desired result. □