I'm trying to prove that, for $k\geq 0$, Sobolev spaces defined in this way:
$H^k(\mathbb{T})=\{f\in L^2(\mathbb{T}): \sum_{n=-\infty}^{+\infty}(1+n^2)^k|\hat{f}(n)|^2 < +\infty\}$
are Hilbert spaces over $\mathbb{C}$ with respect to the inner product:
$(f,g)=\sum_{n=-\infty}^{+\infty}(1+n^2)^k\hat{f}(n)\overline{\hat{g}(n)}$,
where $\hat{f}$ is the Fourier transform of $f$ in $\mathbb{T}=[-\pi, \pi)$.
I proved that $H^k(\mathbb{T})$ is a vector space over $\mathbb{C}$ $\forall k\geq 0$ and that is an inner product space. Now I need to prove that $H^k(\mathbb{T})$ is complete with respect to the distance induced by the norm $||\cdot||=(\cdot,\cdot)^{1/2}$.
So I considered a Cauchy sequence $\{f_m\}_{m\in\mathbb{N}}\subseteq H^k(\mathbb{T})$. This means in particular that, $\forall n\in\mathbb{Z}$, the sequence $\{\hat{f_m}(n)\}_{m\in\mathbb{N}}$ is a Cauchy sequence in $\mathbb{C}$, therefore it converges to some $g(n)\in\mathbb{C}$, because $\mathbb{C}$ is complete. So I defined
$f(x):=\sum_{n=-\infty}^{+\infty} g(n)e^{inx}$, $\forall x\in\mathbb{T}$.
I managed to prove that $\hat{f}(n)=g(n)$, $\forall n\in\mathbb{Z}$, but now I'm finding troubles in showing that $f\in H^k(\mathbb{T})$ and that $f_m\rightarrow f$ with respect to the norm in $H^k(\mathbb{T})$.
Is my idea correct? How could I proceed?
I would personally take the $L^2$ route, to avoid issues I brought up in my above comment. Note that $(f_m)_m$ is Cauchy in $L^2$: for any $\epsilon > 0$ and for $m,l$ large enough, we have $$||f_m-f_l||_2^2 = \sum_{n=-\infty}^\infty |\widehat{f_m}-\widehat{f_l}|^2 \le \sum_{n=-\infty}^\infty (1+n^2)^k |\widehat{f_m}-\widehat{f_l}|^2 \le \epsilon.$$ Since we know $L^2$ is complete, let $f$ be the $L^2$ limit of the $f_m$'s. It remains to show $f \in H^k(\mathbb{T})$ and $f_m \to f$ in $H^k$. But these should just be some easy triangle inequality or Cauchy-Schwarz arguments.