Proposition 10.2.3.label Let $E$ be a vector space over $\complex$ and $*: E \to E$ be a complex conjugation, then:
- (1)
$E = \complex(\text{Re}(E))$.
- (2)
For each $x \in E$,
\[\text{Re}(x) = \frac{x + x^{*}}{2}\quad \text{Im}(x) = \frac{x - x^{*}}{2i}\]
Proof. (2): By properties of the complex conjugation, $\text{Re}(x), \text{Im}(x) \in \text{Re}(E)$.
(1): For any $x, y \in \text{Re}(x)$ with $x = iy$, $x = -iy$ as well by (2) of the complex conjugation, so $x = y = 0$. Thus if $z = x + iy = x' + iy'$, then $x = x'$ and $y = y'$, and the decomposition is unique.$\square$
Post a Comment