How to show this Sobolev embedding?

Question

For $1<p<\infty$, let $W^{1,p}([0,1],\mathbb{R}^n)$ be the periodic Sobolev space of period $1$, i.e. $$ W^{1,p}=\{u \in L^p([0,1],\mathbb{R}^n) \ | \ Du \in L^p([0,1],\mathbb{R}^n)\} $$ with the norm $||u||_{W^{1,p}}=||u||_p+||Du||_p$. How can one prove that the identity map $W^{1,p}([0,1])\rightarrow C([0,1],\mathbb{R}^n)$ is an embedding, where the latter space is endowed with $\infty$-norm?

Answer 1

Suppose $u$ is $C^1$. Let $|u(a)|$ be the minimum of $|u(x)|$ on $[0,1]$. Then $$ || u ||_p = \left(\int |u(x)|^p\right)^{1 \over p} \ge |u(a)|. $$ Let $|u(b)|$ be the maximum of $|u(x)|$ on $[0,1]$. Then $$ ||u||_\infty =|u(b)| \le |u(a)| + \int_a^b |u'(x)| \le ||u||_p + ||u'||_p. $$

Since $C^1$ functions are dense in $W^{1,p}$ this is enough.