Must $\langle x \mid x \rangle$ be real on a complex inner product space?

Question

A complex inner product operator must satisfy four properties, including positivity which says, according to my textbook:

If $x \neq 0$ then $\langle x \mid x \rangle > 0$.

Does this mean that $\langle x \mid x \rangle$ must be real?

Answer 1

Yes, $\langle x, x \rangle$ is always real. This comes from conjugate symmetry, as $\langle x, x \rangle = \overline{\langle x, x \rangle}$.

Answer 2

I think you mean $\langle x,x \rangle$, and indeed it must be real.

Answer 3

It is implied by the skew symmetry of the inner product $$\langle x,y\rangle=\overline{\langle y,x\rangle}$$ applied to $x=y$

$$\langle x,x\rangle=\overline{\langle x,x\rangle}$$