understanding the meaning of scalar in the inner product definition.

Question

The definition is given below:

]1

but I am wondering:

1-In (2) why the author is not putting $\bar{\alpha}$ which means the conjugate of $\alpha$?

2- can the second and the third properties help us to say that the inner product is bilinear? if so how?

Answer 1

1. He defines it that way, it doesn't really make a difference.
2. No, property 4 and 2 tells you it's not bilinear.

If it's bilinear then $\overline{a}<x, y>=\overline{a <y, x>}=\overline{<y, ax>}=<ax, y> = <x, ay> = a<x, y> $ so that $a = \bar{a}$ which is not necessarily true.

Answer 2

From the last three properties we can derive that $$\begin{align} \langle \alpha x,y \rangle &= \overline{\langle y,\alpha x \rangle} \\ &= \overline{\alpha \langle y,x \rangle} \\ &= \overline{\alpha} \overline{\langle y,x \rangle} = \overline{\alpha} \langle x,y \rangle. \end{align}$$ So, in general, we cannot say that $x\mapsto \langle x,y \rangle$ is a linear map, we say instead that is conjugate linear.

Also, observe that if the vector space is real, then $\langle\cdot,\cdot\rangle$ is a bilinear map, because its linear in each of the both entries.