A question about Sobolev functions

Question

Let $\Omega\subset\mathbb{R}^n$ open, bounded and smooth. Let $\lambda_j$ and $e_j$, $j\in\mathbb{N}$, be the eigenvalue and the corresponding eigenfunctions of the Laplacian operator $-\Delta$ in $\Omega$ with zero Dirichlet boundary data on $\partial\Omega$. We suppose that: $|| e_j ||_{L^2(\Omega)}=1$. Let $s\in(0,1)$. Let $u\in H_0^1(\Omega)$, i want to prove that: $$ \sum_{j\in\mathbb{N}}(u,e_j)_{L^2(\Omega)}^2\lambda_j^s<+\infty, $$ where: $$(u,e_j)_{L^2(\Omega)}=\int_\Omega e_ju\,dx. $$ I have no idea to go on, any help would be appreciated.