Studyng of fourier analysis $\widetilde{f:}L^{2}(T)\rightarrow l^{2}(\mathbb{Z})$. The periodic space of Sobolev $H^{s}$ are the functions f in $L^{2}(T)$ such that $\sum_{k\epsilon \mathbb{Z}}(1+\left | k \right |^{2})^{s}\left | \widetilde{f(k)} \right |^{2}< \infty $. how can i show that the next are three norms in $H^{m}$ for $m \epsilon \mathbb{N}$?
$\sum_{j=0}^{m}\left \| f^{(j)} \right \|_{L_{2}}$ with $f'(x)=\sum_{k\in \mathbb{Z}}ik\widetilde{f(k)}e^{ikx}$, $(\sum_{k\epsilon \mathbb{Z}}(1+\left | k \right |^{2s})\left | \widetilde{f(k)} \right |^{2})^{1/2}$ and $(\sum_{k\epsilon \mathbb{Z}}(1+\left | k \right |^{2})^{s}\widetilde{f(k)}\overline{\widetilde{f(k)}})^{1/2}$