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Math Help - Polarization Identity

  1. #1
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    Polarization Identity

    I'm trying to derive the L^2(\mathbb{R} ) inner product using the polarization identity but I'm getting stuck. This is how far I get

    4(f,g) = \Vert f+g\Vert ^2 - \Vert f-g\Vert ^2 <br />
= \int \vert f+g\vert ^2dx -\int \vert f-g \vert ^2dx
    = \int (f+g)(\bar{f} +\bar{g})-(f-g)(\bar{f} -\bar{g})dx<br />
= 2\int \bar{f} g +f\bar{g}dx

    But I want to end up with 4\int f\bar{g} dx. I assume I've gone wrong somewhere???
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  2. #2
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    Quote Originally Posted by markwolfson16900 View Post
    I'm trying to derive the L^2(\mathbb{R} ) inner product using the polarization identity but I'm getting stuck. This is how far I get

    4(f,g) = \Vert f+g\Vert ^2 - \Vert f-g\Vert ^2 <br />
= \int \vert f+g\vert ^2dx -\int \vert f-g \vert ^2dx
    = \int (f+g)(\bar{f} +\bar{g})-(f-g)(\bar{f} -\bar{g})dx<br />
= 2\int \bar{f} g +f\bar{g}dx

    But I want to end up with 4\int f\bar{g} dx. I assume I've gone wrong somewhere???
    I think you're mixing up the real and complex versions of the polarisation identity. In the real case you don't need the bars over the f and g (because they are real!).

    For complex functions you would use the complex polarisation identity 4(f,g) = \|f+g\| ^2 + i\|f+ig\| ^2 - \| f-g\| ^2 - i\|f-ig\| ^2.
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  3. #3
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    Ah of course. Thank you.
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