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Math Help - proof: cauchy-sequence

  1. #1
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    proof: cauchy-sequence

    Hi! (Merry Christmas and happy holidays!! I hope you guys have a good time)

    Exercise

    Proof: (F, ||-||_H) is a Banach space, where

     F := \{f \in L^2(\mathbb{R})  : \int^\infty_{-\infty} (1+x^2)|\hat{f}(x)|^2 dx < \infty \}
    ( \hat{f} fourier transform)

    and  ||-||_H := (\int^\infty_{-\infty}(1+x^2)* \overline{\hat{f}} *\hat{f} dx)^{0.5}

    Definition
    If the norm on F is complete (that is, any Cauchy sequence in F is convergent), then F is called a Banach space

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    According to the Cauchy sequence/norm I'm at a loss what to do.


    Thanks for all your time

    Best regards,
    Rapha.
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  2. #2
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    Quote Originally Posted by Rapha View Post
    Exercise

    Proof: (F, ||-||_H) is a Banach space, where

     F := \{f \in L^2(\mathbb{R})  : \int^\infty_{-\infty} (1+x^2)|\hat{f}(x)|^2 dx < \infty \}
    ( \hat{f} fourier transform)

    and  ||-||_H := (\int^\infty_{-\infty}(1+x^2)* \overline{\hat{f}} *\hat{f} dx)^{0.5}
    Let U be the map from F to L^2(\mathbb{R}) given by (Uf)(x) = \sqrt{1+x^2}\hat{f}(x). Then \|U(f)\|_2 = \|f\|_H. The map U is linear, and it follows that F, with the norm H, is a normed linear space. To show that it is a Banach space, you need to show that U is surjective. The completeness of F then follows from that of L^2(\mathbb{R}).
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