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.

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    $\begingroup$ 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). $\endgroup$
    – LL 3.14
    Sep 22 at 14:22

1 Answer 1


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.


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