The following inequality is named after Otto Hölder (1859 - 1937).

Proposition: Hölder's Inequality

Let $p,q\in(1,\infty)$ with $\frac 1p+\frac 1q=1$. Let $x=(x_1,x_2,\ldots x_n)$ and $y=(y_1,y_2,\ldots y_n)$ be two vectors of a vector space \(V\) over the field of real numbers \(\mathbb R\) or the field of complex numbers \(\mathbb C\). Then the product of the p-norms $||x||_p||y||_q$ can be used as an upper bound of the sum $\sum_{\nu=1}^n|x_\nu y_\nu|$, formally:

$$\sum_{\nu=1}^n|x_\nu y_\nu|\le ||x||_p||x||_q.$$

Proofs: 1

Proofs: 1 2

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  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983