As you may know, there are two ways to define the Sobolev space $W^{k,p}$. One is to define $W^{k,p}$ as the set of all functions whose weak derivatives up to order $k$ are in the Lebesgue space $L^p$. The other is to define $W^{k,p}$ as the completion of a normed space of functions that can be continuously differentiated certain times. The first definition seems rigid and bears little variation in the literature; however, at least two versions of the second definition are conveyed among mathematicians. According to [Adams & Fournier, 2003], $W^{k,p}$ (actually, they use $H^{k,p}$ to distinguish this definition from the one via weak derivatives) is defined as the completion of $C^k$ in the norm $$\lVert u\rVert_{k,p}=\left(\sum_{|\alpha|\leq k}\lVert D^\alpha u\rVert_p^p\right)^\frac{1}{p}.$$ In a later chapter of their book, this definition is proved to be equivalent to the one via weak derivatives, a marvelous result, but recently I saw a definition still in the form of completions but with $C^k$ replaced by $C^\infty$. What's the difference between these two completions? Is it possible to complete two different spaces to the same space? Is the $C^\infty$ definition still equivalent to the one via weak derivatives? Any advice is appreciated. Thank you.
Update: After looking into the proof of the "$H=W$" theorem in [Adams & Fournier, 2003], I guess that these two completions are equivalent. Please see Theorem 3.17 if you have the same question.
Yes, these are equivalent for $p<\infty$ (at least when working with functions defined on the whole space). The only result you need is the fact that $C^\infty_c(\Bbb R^d)$ is dense in $W^{k,p}(\Bbb R^d)$, and so any space in between (i.e. any space verifying $C^\infty_c(\Bbb R^d) \subset X \subset W^{k,p}(\Bbb R^d)$ is also dense in $W^{k,p}(\Bbb R^d)$.
Warning however about the case $p=\infty$ where $C^\infty_c(\Bbb R^d)$ is not dense in $W^{k,\infty}$ (the one defined with weak derivatives). Sometimes (see e.g. Triebel, Theory of functions spaces, or Maz'ya, Sobolev spaces) people write $\mathring{W}^{k,p}(\Bbb R^d)$ to denote the completion of $C^\infty_c(\Bbb R^d)$ with respect to the $W^{k,p}$ norm.
As an example, consider $L^\infty$. Then the completion of $C^\infty_c(\Bbb R^d)$ with respect to the $L^\infty$ norm is $C^0_0(\Bbb R^d)$, the space of continuous functions vanishing at infinity.