# weighted inner product of polynomials, can weight function be complex?

I am just learning about inner-products on polynomial space, where the coefficients of the polynomials may be complex: $P_m(\mathbf{F})$

The inner-product given by:

$\langle p,q \rangle = \int_0^1 p(x) \, \overline{q(x)} dx$

I understand that the inner-product can be weighted:

$\langle p,q \rangle = \int_0^1 r(x) \, p(x) \, \overline{q(x)} dx$

What I don't know (haven't been able to find) is whether the function r(x) has restrictions on it such as "it must be a real-valued function", or strictly positive, or strictly non-negative.

Judging by what I know about norm being non-negative, I think that r(x) can't be negative (otherwise the inner product of a function with it self, i.e., the norm, could be less than 0).

Can anyone help?