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Proposition: Cauchy-Schwarz Inequality for Integral p-norms

Let $[a,b]$ be a closed real interval, and let $f,g:[a,b]\to\mathbb R$ be two Riemann-integrable functions. The for the integral p-norms the Cauchy-Schwarz inequality holds:

$$\int_{a}^b|f(x)g(x)|dx\le \left(\int_{a}^b (f(x))^2dx\right)^{\frac 12}\left(\int_{a}^b (g(x))^2dx\right)^{\frac 12}.$$

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References

Bibliography

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