Let $C_0(\mathbb{R}^n) = \{f :\mathbb{R}^n \rightarrow \mathbb{R}| f \ \mbox{is continuous on} \ \mathbb{R}^n, \lim_{x \rightarrow \infty} f(x) = 0\}$.
I want to characterize the compactness properties for this space.
Some information : 1. $(C_0(\mathbb{R}^n; ||\cdot||_\infty))$ is a complete metric space
- Arzela-Ascoli Theorem characterize compactness in $C(X)$ when $X$ is a compact metric space
Suggestion : the space $\mathbb{R}^n$ is a "one point compactification"
Stereographic projection projection gives one point compactification on $\mathbb{R}^n$, and compactness on $C_0(\mathbb{R}^n)$ can be deduced from compactness on $S^{n-1}$ which Arzela-Ascoli applied (with some help with stereographic projection)
$\textbf{Proposition}$ Let $\mathscr{F}$ be a subset of $C_0(\mathbb{R}^n).$ Assume that $\mathscr{F}$ is closed and bounded (under the norm $||\cdot||_\infty$). Then $\mathscr{F}$ is compact if and only if
(a) for any compact subset $K$ of $\mathbb{R}^n,$ $\mathscr{F}$ is equicontinuous in $\textbf{C(K)}$
(b) for any $\epsilon > 0$, there exists $r > 0$ such that for any $f \in \mathscr{F},$ $$|f(x)| < \epsilon$$ for any $|x|>r.$
I need to prove this. I assume that the suggestion about one point compactification and Stereographic projection will play a crucial role. I do not know about one point compactifiction. Precisely, I know only some basic point-set topology.
Moreover, I do not fully understand the content of the proposition.
- the condition (a); $\mathscr{F}$ is equicontinuous on $C(K)$.
I think $C(K) = \{f : K \rightarrow \mathbb{R}| f \ \mbox{is continuous on} \ K\}$. I feel like $\mathscr{F} \subseteq C_0(\mathbb{R}^n)$ is a set of continuous function with domain $\mathbb{R}^n$.
So what is $\mathscr{F}$ is equicontinuous on $C(K)$ ? The domain seems different.
- condition (b) is like the family $\mathscr{F}$ is uniformly zero at infinity ?
Any suggestions to understand and prove it ?