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

Exercise

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

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

($\displaystyle \hat{f}$ fourier transform)

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

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.