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Proposition: Hölder's Inequality for Integral p-norms
Let $[a,b]$ be a closed real interval, $p,q > 1$ be a real numbers with $1/p+1/q=1,$ and let $f,g:[a,b]\to\mathbb R$ be two Riemann-integrable functions. The for the integral p-norms the Hölder's inequality holds:
$$\int_{a}^b|f(x)g(x)|dx\le ||f||_p||g||_q,\quad\forall p,q > 1:\;\frac 1p+\frac 1q=1.$$
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References
Bibliography
- Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983
