inner product (real or imaginary?)

Question

I have a simple question. Is the inner product $\langle x | z\rangle$ always real or does this hold only for an inner product with itself $\langle x | x \rangle$?

Answer 1

In real vector spaces, both are real, and the latter is also positive.

In complex vector spaces, the first can be anything (e.g. $\left< x|(i | x\right>)$, but the latter is still real and positive.

Answer 2

Hint: Any inner product $\langle {-} | {-} \rangle$ on a complex vector space satisfies $\langle \lambda x | y \rangle = \lambda^* \langle x | y \rangle$ for all $\lambda \in \mathbb{C}$.

You're right in saying that $\langle x|x \rangle$ is always real when the field is defined over the real numbers: in general, $\langle x|y \rangle = \overline{\langle y|x \rangle}$, so $\langle x|x \rangle = \overline{\langle x|x \rangle}$, so $\langle x|x \rangle$ is real. (It's also always positive.)