列紧集
Definitions. $(\mathcal{X},\rho)$ is a normed space.
For $A\in \mathcal{X}$, it is
$\textcolor{darkblue}{\textit{列紧}:}$ 任意点列有收敛子列。
$\textcolor{darkblue}{\textit{自列紧}:}$ 该收敛子列收敛回子集$A$。
$\textcolor{darkblue}{\textit{完全有界}:}$ 任意$\epsilon$都有有穷$\epsilon$网。
$\textcolor{darkblue}{\textit{可分}:}$ 含有可数的稠密子集。
For $C(M)$ and $F\subset C(M)$ where $(M,\rho)$ is acompact normed space, F is
$\textcolor{darkblue}{\textit{一致有界（函数）}:}$ 存在$M_1$, for any $\phi \in F, x \in M, |\phi(x)|\leq M_1$. i.e. subset $F$ is 有界（函数空间） in the normed space $(C(M),d)$, here $d(u,v) = max|u-v|$.
$\textcolor{darkblue}{\textit{等度连续}:}$ For any $\epsilon, \phi \in F$, we have $$\delta(\epsilon)>0,|\phi(x_1)-\phi(x_2)|<\epsilon$$ once $\rho(x_1,x_2)<\delta$.