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July 8th, 2017, 04:12 AM   #1
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Cauchy Schwarz Inequality

Cauchy Schwarz Inequality in Rn: $\displaystyle |x\cdot y|\leq |x||y|$

Proof: $\displaystyle \left | x\cdot \frac{y}{|y|}\right |\leq |x|$

Explanation: $\displaystyle \frac{y}{|y|}$ is a unit vector which can be expanded to a basis. Then either x is in the direction of y and equality holds, or it has a component among the rest of the basis in which case it's component along y is less than |x|.
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July 9th, 2017, 04:18 AM   #2
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