Let $C(\mathbb{R})$ be a Banach space of continuous real-valued functions defined on $\mathbb{R}$, with supremum norm, and let $C_0(\mathbb R)$ be the subspace of functions vanishing at infinity. Is $C_0(\mathbb{R})$ a Banach space?
I try to see it using:
$f\in C_0(\mathbb{R})$ iff for any $\epsilon>0$ there exists $K>0$ such that $|f|<\epsilon$ whenever $|x|>K$. But I think it is not Banach.
Please I need a counter-example or a proof.