For $Q=[-\pi,\pi)$ and $s\geq 0$, define the Sobolev space \begin{equation} H^s(Q):=\{f\in L^2(Q):\sum_{k\in\mathbb{Z}}(1+k^2)^s|\hat f(k)|^2<+\infty\} \end{equation}
- Prove that $H^s(Q)$ is a Hilbert with respect to the inner product \begin{equation} (f,g)_s:=\sum_{k\in\mathbb{Z}}(1+k^2)^s\hat{f}(k)\overline{\hat{g}(k)} \end{equation}
My idea:
Let $\{f_j\}\subset H^s(Q)$ a Cauchy sequence, in particular the sequence is Cauchy in $L^2(Q)$ and then $\exists f\in L^2(Q)$ such that $||f_j-f||_{L^2}\rightarrow 0$ for $j\rightarrow +\infty$. I have to prove that $f\in H^s(Q)$.
I know that: \begin{equation} \hat{f}(k)=\frac{1}{\pi}\int_Qf(x)e^{-ikx}dx \end{equation} so: \begin{equation} \sum_{k\in\mathbb{Z}}(1+k^2)^s|\hat{f}(k)|^2\leq\sum_{k\in\mathbb{Z}}(1+k^2)^s|\frac{1}{\pi}\liminf_{n\rightarrow +\infty}\hat{f_n}(k)|^2 \end{equation} but now I have no ideas to continue.