So I am trying to prove that the following functions $\{f_n=\frac{1}{\sqrt{2 \pi}}e^{inx}\}_{n=-\infty}^{n=\infty}$ are dense in the space $H^{n}[0,2\pi]$. For the proof it would be safe to assume that the functions are dense in $L^2[a,b]$. So to go about proving this I want to use Planceherel's equality i.e. $ \|f\|^2=\sum_{i=-\infty}^\infty (f,f_n)^2$ where $f$ is any function in $H^{n}[0,2\pi]$. now using integration by parts one arrive to the following equality (for the case n=1): \begin{equation} (f,f_n)_{H^1}=(1+n^2)(f,f_n)_{L^2}=(1+n^2)c_n \end{equation} Also we arrive at the equation:
\begin{equation} \|f\|^2=(f,f)=\sum_{-\infty}^{\infty}(1+n^2)c_n\times\bar{c_n} \end{equation} where $c_n=(f,f_n)$. So now I want to prove Plancherel's identity for this case but this is where I am stuck. On one hand you will get a term $(1+n^2)^2$ and on the other just a $(1+n^2)$. What am I doing wrong here?