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Proof

(related to Proposition: Cauchy-Schwarz Inequality for Integral p-norms)

• By hypothesis, $[a,b]$ is a closed real interval and $f,g:[a,b]\to\mathbb R$ are two Riemann-integrable functions.
• By definition of integral p-norms, and since $|y^2|=y^2$ for all $y\in\mathbb R$, $$||f||_2=\left(\int_{a}^b (f(x))^2dx\right)^{\frac 12},\quad ||g||_2=\left(\int_{a}^b (g(x))^2dx\right)^{\frac 12}.$$
• Now, the statement follows as a simple corollary from the Hölder's inequality for integral p-norms for the special case $p=q=2.$
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References

Bibliography

1. Heuser Harro: "Lehrbuch der Analysis, Teil 1", B.G. Teubner Stuttgart, 1994, 11th Edition