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  "body": "\n## Statement\n\nLet $p, q \u003e 1$ be real numbers with $\\frac{1}{p} + \\frac{1}{q} = 1$, and let $a_1, \\dots, a_n, b_1, \\dots, b_n$ be real numbers.\nThen\n$$\\sum_{i=1}^{n} |a_i b_i| \\le \\Big( \\sum_{i=1}^{n} |a_i|^p \\Big)^{1/p} \\Big( \\sum_{i=1}^{n} |b_i|^q \\Big)^{1/q}.$$\n\n## Remarks\n\nScale so that both factors on the right are $1$. Then apply [[Young's inequality|07VHQ31EYJ]] to each product $|a_i| |b_i|$ and add.\nFor $p = q = 2$ it is the Cauchy–Schwarz inequality.\nMathlib states a form as `Real.inner_le_Lp_mul_Lq`.\n\n## Explanations\n\n### L1 human\n\nHölder's inequality bounds a sum of products by the sizes of the two sequences.\nEach size is measured with its own exponent, $p$ for one and $q$ for the other.\n",
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