Question about the complex inner product axioms

My textbook claims that from the axioms for the complex inner product:

$$\left<y,x\right>=\overline{\left<x,y\right>}\tag{1}$$ $$c\left<x,y\right> = \left<cx,y\right>\tag{2}$$

we can derive:

\begin{align} \left<x,cy\right> &= \overline{\left<cy,x\right>}\\ &= \overline{c}\overline{\left<y,x\right>}\\ &= \overline{c}\left<x,y\right> \end{align}

I understand the first and last steps of the derivation, but in the middle step, I don't understand what justifies bringing the $c$ out of the inner product and taking its complex conjugate.

• Should (1) read $\left<x,y\right>=\overline{\left<y,x\right>}$ ? – Henry May 22 '17 at 3:06

$\langle cy, x\rangle = c\langle y,x\rangle$ by the second rule, and so $\overline{\langle cy, x \rangle} = \overline{c\langle y,x \rangle} = \overline{c}\overline{\langle y,x\rangle}$. That last equality just follows from the fact that for any $z,w\in \mathbb C$, we have $\overline{z\cdot w} = \overline{z}\cdot\overline{w}$.
The definition of an inner product has that its sesquilinear -- linear in one of the arguments, conjugate linear in the other argument. The pulling $c$ out with the conjugate is part of the sesquilinearity.