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I have a question of the algebra in the triangle inequality with complex numbers. First, is it correct to write$$ z\overline{w}+w\overline{z}=z\overline{w}+\overline{z\overline{w}}=2z\overline{w}=2w\overline{z} $$ And the main question, why can we write $$ z\overline{w}+w\overline{z}=2\operatorname{Re}\{{z\overline{w}}\} $$ And thereby, why is $$ 2\operatorname{Re}\{{z\overline{w}}\} \leq 2 \left |z\overline{w} \right |= 2 \left |z \right | \left |w \right | $$

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First, is it correct to write$$ z\overline{w}+w\overline{z}=z\overline{w}+\overline{z\overline{w}} =\color{red}{2z\overline{w}=2w\overline{z}} $$

You were doing fine until you reached the parts in red. Those are errors. Next step after $z\bar{w} + \overline{z\bar{w}}$ should be $\color{blue}{= 2\operatorname{Re}(z\overline{w})}$.

And the main question, why can we write $$ z\overline{w}+w\overline{z}=2\operatorname{Re}\{{z\overline{w}}\} $$

Remember, given a complex number $c$, $c + \bar{c} = 2\operatorname{Re}(c)$. This is because if $c = a + bi$, then $\bar{c} = a - bi$ and thus $c + \bar{c} = 2a = 2\operatorname{Re}(c)$.

And thereby, why is $$ 2\operatorname{Re}\{{z\overline{w}}\} \leq 2 \left |z\overline{w} \right |= 2 \left |z \right | \left |w \right | $$

Note $a\le \lvert a\rvert = \sqrt{a^2} \le \sqrt{a^2 + b^2}$, thus $\operatorname{Re}(c) \le \lvert c\rvert$. Taking $z\overline{w}$ for $c$, you get $\operatorname{Re}(z\overline{w}) \le \lvert z\overline{w}\rvert$. That should help you to see the first inequality. The second inequality follows from $\lvert z\overline{w}\rvert = \lvert z\rvert \lvert \overline{w}\rvert$ and $\lvert \overline{w}\rvert = \lvert w\rvert$.

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let $z = x + iy, w = a+ib$

$z\bar w = (x+iy)(a-ib) = xa + yb + i(-xb + ya)\\ \bar z w = (x-iy)(a+ib) = xa + yb + i(xb - ya)$

And hopefully you can see that the two expressions are not the same.

The real parts are the same. The imaginary parts have opposite signs.

$Re\{z\bar w\} = Re\{\bar z w\}$ and $Im\{z\bar w\} = - Im\{\bar z w\}$

$z\bar w+ w\bar z \ne 2z\bar w$.

$z\bar w+ w\bar z = (Re\{z\bar w\} + Re\{\bar z w\}) + i (Im\{z\bar w\} + Im\{\bar z w\}) = 2Re\{z\bar w\}$

Hope this helps.

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