I've read a paper, and in the proof of Theorem 2, the authors cited a theorem of Ascoli (as the second theorem of Ascoli). I'd like to see the full statement of the cited theorem, but the reference in the paper is written in (maybe) French so I cannot demonstrate it.
Can anybody help me?
Is it true, and can I find a reference (written in English)?: pointwise convergence of real-valued equicontinuous function sequence $(f_j)_{j \geq 1}$ defined on $X$ to a target function $f$ implies that uniform convergence to the same function on the closure of $X$.
Here is a translation of the theorem cited in Laurent Schwartz's book :
Theorem : Let $E$ be a topological space and $F$ a semi-metric space. On an equicontinuous set $\mathcal{F}$ of functions from $E$ to $F$, the semi-metric structures of the pointwise convergence on a dense subset $E_0 \subset E$, of the pointwise convergence on $E$, and of the uniform convergence on every compact of $E$, are uniformly equivalent (which means that the corresponding uniform structures are the same, and in particular the corresponding topologies are the same).
(and because it is written by Laurent Schwartz, I guess you can assume it is true...)