Compactness in $C_0(\mathbb{R})$

Question

Is there a compact set in $C_0(\mathbb{R})$ (continuous functions vanishing at infinity) that contains the unit sphere of $C_0^1(\mathbb{R})$ (differentiable functions in $C_0(\mathbb{R})$ such that the derivative is also in $C_0(\mathbb{R})$)? The norm in the Banach space $C_0^1(\mathbb{R})$ being defined as $\|f\|_1:=\max(\|f\|,\|f'\|)$.

Answer 1

Let $\phi$ a smooth function with support in $[0,1]$, and $\phi=1$ on $(1/4,3/4)$. Consider the sequence $f_n(x):=\frac{\phi(x+n)}{\lVert \phi\rVert+\lVert \phi'\rVert}$. Then $\{f_n\}$ is a sequence which lies in the unit ball of $C^1_0$. But $\lVert f_m-f_n\rVert_{\infty}=\frac 1{\lVert f\rVert+\lVert f'\rVert}$, so we can't find a compact set $K$ of $C_0(\Bbb R)$ containing the unit ball of $C^1_0(\Bbb R)$.