I am learning about the Sobolev spaces from the book "Theoretical numerical analysis : a functional analysis framework" by Atkinson and Ham.
Here is the excerpt from the book that I do not understand:
I do not understand the part that says "Since $C^\infty(\overline\Omega) \subset C^k(\overline\Omega) \subset W^{k,p}(\Omega) $". Why do we need to say anything about the space "$C^k(\overline\Omega)$" ?
Isn't it sufficient to say just that"Since $C^\infty(\overline\Omega) \subset W^{k,p}(\Omega) $. "?
The author just wanted to be more clear about the reason for inclusion:
$u\in C^\infty(\overline{\Omega}) \implies $ $u$ has continuous classical derivatives of orders up to $k$ $\implies $ $u$ has weak derivatives of orders up to $k$, which are in $L^p$.
The implications are expressed as inclusions, the middle step being $u\in C^k(\overline{\Omega})$.