= Solution
Assume the intended nondegenerate <interval> $a<b$ and real-valued <continuous functions>. The integral is bilinear and symmetric. Since $w>0$, it gives $\langle f,f\rangle\ge0$. If a <continuous> $f$ is nonzero at some point, <continuity> makes $f^2w$ strictly positive on a subinterval of positive length, so its integral is positive. Hence $\langle f,f\rangle=0$ forces $f=0$, proving positive definiteness. <Orthogonality> means $\boxed{\int_a^bf(x)g(x)w(x)dx=0}$. 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.
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