Exercise 6.1.30

First we may observe that the condition for norm on real vector space is loosen than that on complex vector space. So naturally the function $\text{∥⋅∥}$ is still a norm when we regard $V$ as a vector space over $ℝ$. By Exercise 6.1.27, we’ve already defined a real inner product $\text{[⋅,⋅]}$ on it since the parallelogram law also holds on it. And we also have

So by Exercise 6.1.29 we get the desired conclusion.