Exercise 6.A.24

Proof. For positivity, we have ${⟨v,v⟩}_1=⟨\mathit{Sv},\mathit{Sv}⟩\ge 0$.

For definiteness, suppose ${⟨v,v⟩}_1=0$. Then $⟨\mathit{Sv},\mathit{Sv}⟩=0$, which implies that $\mathit{Sv}=0$. However $S$ is injective, thus $v=0$. Conversely, if $v=0$, then ${⟨v,v⟩}_1=⟨\mathit{Sv},\mathit{Sv}⟩=⟨0,0⟩=0$.

For additivity in the first slot, we have ${⟨u+v,w⟩}_1=⟨\mathit{Su}+\mathit{Sv},\mathit{Sw}⟩=⟨\mathit{Su},\mathit{Sw}⟩+⟨\mathit{Sv},\mathit{Sw}⟩={⟨u,w⟩}_1+{⟨v,w⟩}_1$.

For homogeneity, ${⟨\mathit{\lambda u},v⟩}_1=⟨\mathit{\lambda Su},\mathit{Sv}⟩=\lambda ⟨\mathit{Su},\mathit{Sv}⟩=\lambda {⟨u,v⟩}_1$.

For conjugate symmetry, we have ${⟨u,v⟩}_1=⟨\mathit{Su},\mathit{Sv}⟩=\overline{⟨\mathit{Sv},\mathit{Su}⟩}=\overline{{⟨v,u⟩}_1}$. □