Past exam of the Probability, statistics and options II course of the University of Oxford / 2010 / P / P4 / a / Solution

Assume the intended nondegenerate interval $a<b$ and real-valued continuous functions. The integral is bilinear and symmetric. Since $w>0$, it gives $⟨f,f⟩\ge 0$. If a continuous $f$ is nonzero at some point, continuity makes $f^{2}w$ strictly positive on a subinterval of positive length, so its integral is positive. Hence $⟨f,f⟩=0$ forces $f=0$, proving positive definiteness. Orthogonality means $\boxed{{\int }_a^{b}f(x)g(x)w(x)dx=0}$. If $a=b$, the displayed integral is identically zero and is not an inner product on the stated function space; that degenerate case must be excluded.