I'm currently reading Lions and Magenes Non-Homogeneous Boundary Value Problems and Applications and I'm stuck at one point.
Let $\Gamma$ be the smooth boundary of an open bounded subset $\Omega\subset\mathbb{R}^n$. We can define the Laplace-Beltrami operator $\Delta$. Denote the eigenvalues of $-\Delta$ by $\lambda_j$ and the orthonormalized (in $L^2(\Gamma)$) eigenfunctions by $w_j$.
At the beginning of the book they use local charts to define the spaces $H^s(\Gamma)$ for $s\in\mathbb{R}$. Later they state that we can equivalently define $H^{2m}(\Gamma)$ for $m\geq 0$ by $$H^{2m}(\Gamma)=\lbrace u: u\in L^2(\Gamma), \ \Delta_\Gamma^m u\in L^2(\Gamma)\rbrace$$
with equivalent norm $\Vert u\Vert_{L^2(\Gamma)}+\Vert\Delta_\Gamma^m u\Vert_{L^2(\Gamma)}$. Then they say by interpolation we can show for $s\in\mathbb{R}$ $$H^s(\Gamma)=\lbrace u: u\in\mathcal{D}'(\Gamma) \ \sum_{j=1}^\infty \lambda_j^{2s}\vert \langle u,w_j\rangle\vert^2<\infty\rbrace.$$
What puzzles me, or rather what I wonder about:
Is the norm on $H^s$ given by $\left(\sum_{j=1}^\infty \lambda_j^{2s}\vert \langle u,w_j\rangle\vert^2\right)^\frac{1}{2}$? Looking at the case $s=2m$ and the norm of the graph, I would think that it should be something like $\left(\sum_{j=1}^\infty (1+\lambda_j^{2})^s\vert \langle u,w_j\rangle\vert^2\right)^\frac{1}{2}$ for $s\geq 0$? But what about $s<0$? What can we say about $\lambda_j$?