# necessary and sufficient conditions for the existence of solution in the space $W^{k,p}$

I am learning about weak derivatives and sobolev space. In particular I need help to learn the proving strategy/technique.

I have trouble proving on how to show a solution belongs to some sobolev space. A particular problem I came across is to prove $$u\in W^{k,p}(\mathbb{R}^d)$$ if and only if $\exists$ a sequence of functions $\{u_m\}\subset C^{\infty}(\mathbb{R}^d)\cap L^p (\mathbb{R}^d)$ such that

1. $$\|u_m-u\|_{L^p(\mathbb{R}^d)}\to 0\quad\text{as}\quad m\to\infty$$ and
2. $$\|D^{\alpha}u_m-D^{\alpha}u_n\|_{L^p(\mathbb{R}^d)}\to 0\quad\text{as}\quad m,n\to\infty\quad\text{for each}\quad |\alpha|\le k$$

Questions Could anyone sketch a proof of above? What books contain proofs of this kind? So that I can pick up the proving technique quickly...

• I think this is not about a criterion for testing whether something is in a given Sobolev space, but, rather, the assertion (for you to prove) that smooth functions (in fact, test functions) are dense in every Sobolev space. – paul garrett Dec 8 '15 at 0:33
• @paulgarrett: On"smooth functions are dense in every sobolev space". How do I show this? Or in what book can I look this up? – math101 Dec 8 '15 at 4:59

EDIT: Broad strokes of iff proof: Forward direction: If $u\in W^{k,p}$, then we have a sequence of smooth functions satisfying 1.,2. by the global approximation theorem. For the reverse direction, if we have a sequence $u_m$ for which $D^\alpha u_m\rightarrow D^\alpha u$ for each $\alpha$, then by completeness of $W^{k,p}$ it must be that $u\in W^{k,p}$.
• Only one direction of the iff is nontrivial: $W^{k,p}$ is complete (a separate, but nontrivial important result) so any sequence in $W^{k,p}$ that converges will be in $W^{k,p}$... Thus the other direction is where the only work happens. – charlestoncrabb Dec 8 '15 at 1:46
• I cannot understand what you mean? 1) which direction ? 2) How does this "iff" type of proof relates to the the sentence you wrote $W^{k,p}$ is complete? Do you mean...the proof of $u\in W^{k,p}$ is done if we can show the limit of $u_m$ and the limit of $D^{\alpha} u_m$ can be found in $L^p?$ – math101 Dec 8 '15 at 3:53
• Also, on "completeness of $W^{k,p}$, do you mean completeness of $L^p$? – math101 Dec 8 '15 at 6:14