Show that (mathcal{C}_{infty}:=left{u: mathbb{R}^{d} ightarrow mathbb{R}ight.) : continuous and (left.lim _{|x| ightarrow infty} u(x)=0ight}) equipped with the

Question:

Show that $\mathcal{C}_{\infty}:=\left\{u: \mathbb{R}^{d} ightarrow \mathbb{R}ight.$ : continuous and $\left.\lim _{|x| ightarrow \infty} u(x)=0ight\}$ equipped with the uniform topology is a Banach space. Show that in this topology $\mathcal{C}_{\infty}$ is the closure of $\mathcal{C}_{c}$, i.e. the family of continuous functions with compact support.

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