Exercise 7.C.10

Proof. Suppose (a) holds, so $S$ is an isometry. Then 7.42 ("(a) $\Rightarrow$ (f)") implies that $S{S}^{\text{∗}}=I$. Therefore, for all $u,v\in V$ we have

Thus (b) holds. Clearly (b) implies (c) and (c) implies (d). Now suppose (d) holds. Then 7.42 ("(d) $\Rightarrow$ (g)" with $S$ replaced with $S^{\text{∗}}$) implies that $S$ is an isometry. Thus (a) holds. □