Compact embedding

Question

Prove that the embedding

$j\colon (C^1[0,1],\|\cdot\|)\to(L^1[0,1],\|\cdot\|_{L^1})$

where

$\|f\|=\max\{\|f\|_\infty,\|f'\|_\infty\}$ and $\|f\|_\infty$ denotes the supremum norm,

$\|f\|_{L^1}=\int_0^1|f(t)|dt$,

is compact.

I could prove it with the Arzelà-Ascoli, but I don't know if that theorem only works when you want to prove the compactness of a set $M\subset C(S)$, where $S$ is compact.

Could I use Arzelà-Ascoli? Otherwise, how can I solve the problem?

Answer 1

Yes, you can use Ascoli-Arzelà. Most easily seen here if you factor the map through $L^\infty([0,1], \lVert\,\cdot\,\rVert_{L^\infty})$:

$$C^1([0,1],\lVert\,\cdot\,\rVert) \underbrace{\hookrightarrow}_{\text{compact}} L^\infty([0,1], \lVert\,\cdot\,\rVert_{L^\infty}) \underbrace{\hookrightarrow}_{\text{continuous}} L^1([0,1], \lVert\,\cdot\,\rVert_{L^1}).$$