I am looking at the proof of the following lemma and I don't understand the conclusion.
Lemma: For $k\geq 0$ integer, let $f=\sum_{n\in \mathbb{Z^d}}{c_n\chi_n}\in L^2(\mathbb{T}^d)$. Then $$f\in H^k(\mathbb{T}^d)\iff \sum_{n\in \mathbb{Z}^d}{\mid c_n\mid^2\parallel n\parallel_2^{2k}}<\infty.$$
Proof of "$\Leftarrow$": For any $\alpha :\parallel\alpha\parallel_1\leq k$ we have $n_1^{\alpha_1}\cdots n_d^{\alpha_d}=n^\alpha\leq\parallel n\parallel_2^{k}$. So the sum $$\sum_{\mid\alpha\mid\leq k}{\sum_{n\in \mathbb{Z}^d}{\mid(2\pi in)^\alpha c_n\mid^2}}<\infty.$$
From this it should follow that the partial sums $\sum_{\parallel n\parallel_2<N}{c_n\chi_n}$ of the Fourier series of $f$ converges even in the $H^k$ norm induced from $L^2(\mathbb{T})^{K(k)}$.
I think I need more details. I don't get why the convergence of the series above should imply the convergence of the Fourier partial sums w.r.to the $H^k$ norm. Thanks for any help.