How to prove the compactness of this Sobolev embedding?

Question

I have a question on compactness of the following Sobolev embedding.

Let $W^{1,p}([0,1],\mathbb{R}^n)$ be the Sobolev space of functions $u:[0,1]\rightarrow \mathbb{R}^n$ for $1<p<\infty$. How can one show that the identity map $$ W^{1,p}([0,1])\rightarrow C([0,1],\mathbb{R}^n), $$ which turns out to be embedding, is compact, where the latter space is endowed with $\infty$-norm?

Answer 1

Hint: Try to do it for $n=1$ first. Apply the Arzelà–Ascoli theorem. You may want to use the density of $C^1$ functions in $W^{1,p}[0,1]$.