Definition 10.2.2 (Complex Conjugation).label Let $E$ be a vector space over $\complex$ and $*: E \to E$ be a $\real$-linear map, then $*$ is a complex conjugation if:

1. (1)

   For each $\lambda \in \complex$, $(\lambda x)^{*} = \ol \lambda x^{*}$.

2. (2)

   For each $x \in E$, $x^{**}= x$.

In which case, $\text{Re}(E) = \bracs{x \in E| x^* = x}$ is the real part of $E$.