Cauchy–Schwarz inequality
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For the inner product space of square-integrable complex-valued functions, the following inequality holds. | ∫ R n f (x) g (x) ¯ d x | 2 ≤ ∫ R n | f (x) | 2 d x ∫ R n | g (x) | 2 d x. {\displaystyle \left|\int _{\mathbb {R} ^{n}}f(x){\overline {g(x)}}\,dx\right|^{2}\leq \int _{\mathbb {R} ^{n}}|f(x)|^{2}\,dx\int _{\mathbb {R} ^{n}}|g(x)|^{2}\,dx.}

The Hölder inequality is a generalization of this.
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