# Sobolev space $W^{k,p}$ as the completion of $C^\infty$ or the completion of $C^k$

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.

• I would rather say "there are several ways" instead of "there are two ways" ... you might not know other ways ... I would personally define $W^{k,p}$ as the space of distributions such that all the derivatives up to order $k$ in the sense of distributions are in $L^p$ (very close but formally different from weak derivatives). Sep 22 at 14:22

## 1 Answer

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.