(a) Define the Sobolev space $H^{s}\left(\mathbb{R}^{n}\right)$ for $s \in \mathbb{R}$.

(b) Let $k$ be a non-negative integer and let $s>k+\frac{n}{2}$. Show that if $u \in H^{s}\left(\mathbb{R}^{n}\right)$ then there exists $u^{*} \in C^{k}\left(\mathbb{R}^{n}\right)$ with $u=u^{*}$ almost everywhere.

(c) Show that if $f \in H^{s}\left(\mathbb{R}^{n}\right)$ for some $s \in \mathbb{R}$, there exists a unique $u \in H^{s+4}\left(\mathbb{R}^{n}\right)$ which solves:

$$\Delta \Delta u+\Delta u+u=f$$

in a distributional sense. Prove that there exists a constant $C>0$, independent of $f$, such that:

$$\|u\|_{H^{s+4}} \leqslant C\|f\|_{H^{s}}$$

For which $s$ will $u$ be a classical solution?