We define the Sobolev-Space $H^k(\Omega)$ as $\{ f \in L^2(\Omega)\ | \ \|f\|_{H^k(\Omega)}=\left(\sum_{|\alpha|\leq k} \|\partial^{\alpha}f\|_{L^2(\Omega)}^2\right)^{1/2}<\infty\}$ where $\partial^\alpha$ denotes the weak partial derivative. This is again a Hilbert Space (w.r.t $\|.\|_{H^k(\Omega)}$), therefore its Dual Space, denoted by $H^{-k}(\Omega)$, is isomorphic to $H^k(\Omega)$ due to Riesz Represenation Theorem. My problem is now to understand the Inclusions given in our lecture notes:
- $H^k\subset\ ... \subset H^{1} \subset L^2 \subset H^{-1} \subset\ ...\subset H^{-k}$
- Or in particular: $V\subset L^2 \subset V^*$
for $V=H^1_0(\Omega)$, the completion of $C^\infty_c(\Omega)$ w.r.t. $\|.\|_{H^1(\Omega)}$
Another question would be, if it is true, that $H^{-k}(\Omega)$ consists of all the k-th order distributional derivatives of $L^2(\Omega)$ functions?