Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 3 / 19C / a / Solution

Bilinearity and symmetry follow from the corresponding properties of the integral. Since $w(x)>0$,If this integral is zero, the nonnegative continuous integrand must vanish everywhere, so $f=0$. Therefore the displayed formula is a real inner product on $C[a,b]$.