Given $\frac{1}{p} + \frac{1}{q} = 1 $, let $S = f \in L^p(\mathbb{R})$ $spt(f) \subset [-1,1]$, and $\|f\|_p \leq 1$ , and let $ g$ be a fixed but arbitrary function in $ L^1(\mathbb{R})$, with spt(g) \subset [-1, 1]. Show that the image of S under the map $ f \mapsto f * g$ is a compact set in $ C_0([-2, 2])$.
Sketch of my proof:
I know that in a finite dimensional space it suffice to prove the image of S bounded and closed. However, the space of continuous functions is infinite dimensional vector space with basis ${1,x,x^2,...}$. Does that fact that $C_0$ is a metric space dive us any benefit?