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Is there a generalization of the Cauchy-Schwarz Inequality for multiple integrals?

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    You mean like $|\iint f\overline g|\leq\sqrt{\iint|f|^2}\sqrt{\iint|g|^2}$ on subsets of $\mathbb{R}^2$, and higher dimensional analogues? If so, then yes, these are all special cases of the Cauchy-Schwarz inequality: http://en.wikipedia.org/wiki/Cauchy-schwarz2011-04-04

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