Exercise 6.C.6

Proof. Suppose $P_U{P}_W=0$. Then $\mathrm{range}<!--\; FUNCTION\; APPLICATION\; -->P_W\subset \mathrm{null}<!--\; FUNCTION\; APPLICATION\; -->P_U$. However, 6.55 shows that $\mathrm{range}<!--\; FUNCTION\; APPLICATION\; -->P_W=W$ and $\mathrm{null}<!--\; FUNCTION\; APPLICATION\; -->P_U=U^{\text{⊥}}$. Thus $W\subset U^{\text{⊥}}$, which implies that $⟨u,w⟩=0$ for all $u\in U$ and all $w\in W$.

Conversely, suppose $⟨u,w⟩=0$ for all $u\in U$ and all $w\in W$. It follows that $W\subset U^{\text{⊥}}$. Let $v\in V$. By 6.47, we can write $v=w+u$, where $w\in W$ and $u\in W^{\text{⊥}}$. Then

Where the last equality, follows from the fact that $W\subset U^{\text{⊥}}$. □