property used in this inner product equality

Question

For a vectors space over a field F, the length of a vector $\vec{u}$ is

$\mid\mid\vec{u}\mid\mid^{2}=\langle\vec{u},\vec{u}\rangle=\sum_\left(i=1\right)^n\langle\vec{u},\vec{e}_{i}\rangle\langle\vec{e}_{i},\vec{u}=\sum_\left(i=1\right)^n \mid\langle\vec{u},\vec{e}_{i}\rangle\mid^{2}$

Can someone explain to me how the last equality came to be? is there a theorem or property that I'm not recalling?

Thanks in advance.

Answer 1

Since $\langle \vec e_i, \vec u \rangle = \overline{\langle \vec u, \vec e_i \rangle}$ we have $\langle \vec u, \vec e_i \rangle \langle \vec e_i, \vec u \rangle = \langle \vec u, \vec e_i \rangle \overline{\langle \vec u, \vec e_i \rangle} = |\langle \vec u, \vec e_i \rangle|^2$.