What (steps) do I need to show to show that Sobolev spaces
$$H_k(\Omega)$$
for all $k \in \mathbb{N}_0$
are Hilbert spaces?
Sobolev space:
Sobolev space $H_k (Ω)$ consists of all functions $u ∈ L_2 (Ω)$ for which the weak partial derivative $∂^α u$ is in $L_2(Ω)$ whenever $α ∈ N_0^n$ and $|α| ≤ k$.
Additionally some definitions may equip this space with an inner product and a norm.