Let $f\in H^s[0,2\pi]$, be Sobolev space of periodic functions then we have the following norm $$ \Vert f\Vert_{s}=\{ \sum_{m\in \mathbb{Z}} (1+m^2)^s\vert \widehat{f_m}\vert^2 \}^{1/2}$$ (Given in Linear Integral equation by Rainer Kress ch. 8) Further the standard sobolev norm would be $$\Vert f\Vert_{s}= \{\sum_{k\leq s}\Vert D^{(k)}f\Vert_{L^2([0,2\pi])}^2\}^{1/2}.$$
The benefit of the first definition over the second is that it can be defined for $s\geq 0$, whereas the second requires $s$ to be an integer.
Can anyone please provide me with some hint on how to show these both norms are equivalent when $s$ is an integer? Thanks.