I am reading about Sobolev spaces and I have a question regarding Sobolev spaces and the Fourier transform.
So by defining the Fourier transform, $F(\cdot)$ as an isometry we get $||f||_{L^2}=||\widehat{f}||_{L^2}$ and $\widehat{D^\alpha f}=\xi^\alpha\widehat{f}$. So one can define the norm in $H^k$, $||f||_{H^k}=\left(\int(1+|\xi|^2)^k|\widehat{f(\xi)}|^2d\xi\right)^{\frac{1}{2}}$ then he goes on and says that the proof that this is an equivalent norm is an easy exercise and is ommited. So I tried to prove it but I got stuck very quickly. The proof boils down to having to show that the functions $F(\xi)=(1+|\xi|^2)^k$ and $S(\xi)=\sum_{|\alpha|\le k}|\xi^\alpha|^2$ with $\xi\in \mathbb{R}^n$ bound each other. The one directions is more or less clear since $S$ includes the terms of $F$, but I cannot see why the reverse is true.