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

Question

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...

Answer 1

Almost all graduate-level PDE texts will have this classical result, my personal favorite is Evans'. If you just google "global approximation of smooth functions", any number of proofs will pop up, and here was a previous post in which a user had questions about Evans' proof.

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 $u\in W^{k,p}$.