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In $\mathbb{C}^2$, I want to show that $\langle x,y \rangle =xAy^*$ is an inner product space. I almost done with other properties of inner product but left only one property: $\langle x,x \rangle >0$ if $x \neq 0$.

To prove this,
$$\begin{align*} \langle x,x \rangle&=(x_1, x_2)A(x_1, x_2)^*\\ &=\|x_1 \|^2-ix_2\overline{x_1}+ix_1\overline{x_2}+\|x_2 \|^2\\ &=\|x_1 \|^2 +2\operatorname{Re}(ix_1\overline{x_2})+\|x_2 \|^2 \end{align*}$$ where $\operatorname{Re}(z)$ is the real part of the complex number $z$.

I can't follow the last equality. Why does $\operatorname{Re}(z)$ appear here?

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For any complex number $w$, we know that $w + \bar w = 2 \Re(w)$.

Apply this with $w = ix_1\bar x_2$.

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