Exercise 2.3.17

If $T=T^2$, then we have $V=\{y:T(y)=y\}+N(T)$ since $x=T(x)+(x-T(x))$ and we have $T(T(x))=T(x)$ and $T(x-T(x))=T(x)-T(x)=0$. On the other hand, we have if $x\in \{y:T(y)=y\}\cap N(T)$ then we also have $x=T(x)=0$. So by arguments above we have $V=\{y:T(y)=y\}\oplus N(T)$. Finally we have that $T$ must be the projection on $W_1$ along $W_2$ for some $W_1$ and $W_2$ such that $W_1\oplus W_2=V$.