1
$\begingroup$

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?

$\endgroup$
4
$\begingroup$

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

$\endgroup$
4
$\begingroup$

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

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

$\endgroup$
2
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.