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    Integration: Cauchy-Schwarz Inequality

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    Suppose that the functions g:[a,b]-> R are continuous. Prove that:

    The integral from a to b of gf <= (the square root of the integral from a to b of g^2) multiplied by (the square root from a to b of f^2)

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    Please see the attached file for the complete solution.
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    We want to show that:

    A good method of considering this problem is to narrow it down to an inner product problem. First we must show that theses integrals actually represent inner products:

    An inner product should have the following properties:

    1- <x, x> >= 0 and <x, x> = 0 if and only if x=0
    2- <y, x> = <y, x>
    3- <cx, y> = c<x, y>
    4- <x+y, z> = <x, z> + <y, z>

    I just show you the first feature in this ...

    Solution Summary

    The Cauchy-Schwarz inequality is used to prove an integral relation.