Why is $\langle y,x\rangle +\langle x,y\rangle=2\Re \langle x,y\rangle$?

Question

I wonder about the second step of the proof shown below (d) in the picture attached. Why is $\langle y,x\rangle +\langle x,y\rangle=2\Re \langle x,y\rangle$?

Answer 1

For a hermitian inner product we have $$\langle y,x\rangle=\overline{\langle x,y\rangle}$$ so

$$\langle y,x\rangle+\langle x,y\rangle=\overline{\langle x,y\rangle}+\langle x,y\rangle=2\operatorname{Re}\langle x,y\rangle$$