It's a good exercise to show that the following is an inner product for $f$ and $g$ continuous functions:
$$\langle f,g\rangle=\int_a^b f(t)g(t)\, dt\qquad f,g\in \mathcal C[a,b]$$
Hence the integral inequality $$\left|\int_a^b f(t)g(t)\,dt\right|\leq \left(\int_a^b f^2(t)\,dt\right)^{1/2}\left(\int_a^b g^2(t)\,dt\right)^{1/2}$$ immediately follows from the inner product-norm inequality: $|\langle f,g\rangle| \leq \|f\|\|g\|$.
However if we're dealing with $f$ and $g$ Riemann integrable functions then we can't show that we're dealing with an inner-product since we loose the definiteness property (e.g. take $f\equiv 0$ on all $[a,b]$ except at some point $x_0$ we have $f=1$).
But in the proof of that inequality do we require definiteness? (In which case the result immediately applies)
Best Answer
Such a "product" defined on Riemann-integrable functions is still semi-definite, i.e. $\langle f,\,f\rangle$ as defined would never be negative, but it might be $0$ for non-vanishing $f$. (See also here & here.) So while we can prove the inequality, we can't show it's only saturated for linearly dependent $f,\,g$.