Let $\{u_i\}_{i\in \mathcal{I}}\subset C([0,T];\mathcal{S}'(\mathbb{R}^n))$ be a family of continuous maps from $[0,T]$ to $\mathcal{S}'(\mathbb{R}^n)$. Suppose that $\{u_i\}_{i\in \mathcal{I}}$ is equicontinuous on $[0,T]$, and that $\{u_i\}_{i\in \mathcal{I}}$ is uniformly bounded. Can we conclude that there exists a subsequence $u_n \to w\in C([0,T];\mathcal{S}'(\mathbb{R}^n))$?
This seems straightforward if we replace $\mathcal{S}'(\mathbb{R}^n)$ by some metric space and then it's just the Arzelà–Ascoli theorem. Is there a counterpart of the Arzelà–Ascoli theorem for general topological space?
Any hint will be appreciated. Thanks!