I was read a detailed proof of Picard–Lindelöf theorem. And, in this article of Wikipedia I found a notation some weird. I saw that: $$\overline {I_{a}(t_{0})} = [t_0-a,t_0+a]$$ $$\overline {B_{b}(y_{0})} = [y_0-b,y_0+b]$$ This is al normal, it is a definition. But after, I saw:
$$\displaystyle \Gamma : {\mathcal {C}} \ ( \ I_{a}(t_{0}) \ , \ B_{b}(y_{0}) \ )\longrightarrow {\mathcal {C}} \ ( \ I_{a}(t_{0}) \ , \ B_{b}(y_{0}) \ )$$
So, What does this line over the letter mean? It is an error? Because I had the idea that it is:
$$\displaystyle \Gamma : {\mathcal {C}} \ \left( \ \overline {I_{a}(t_{0})} \ , \ \overline {B_{b}(y_{0})} \ \right)\longrightarrow {\mathcal {C}} \ \left( \ \overline {I_{a}(t_{0})} \ , \ \overline {B_{b}(y_{0})} \ \right)$$
The other thing is, according to me, the theorem should have a proof with $\overline {B_{b}(y_{0})} \subseteq \mathbb{R}^n$, but $\overline {B_{b}(y_{0})}$ is a interval in this proof.
Sorry for my english.
Yes, the wikipedia entry is sub-optimal. There should be mention of a domain $\cal D$ of $f$ in the first statement in the intro, later that the cylinder $\bar I_a\times \bar B_b$ has to be a sub-set of $\cal D$.
And you need the closed sets, first for compactness to have a maximum $M$ of $f$ over the cylinder and later for completeness of the function space as you apply the Banach fixed-point theorem.
(Btw., there is no need to invoke Grönwall, as the BFS already supplies the uniqueness of the fixed point.)
In short, as always with the things written in wikipedia, look for other sources (more than one). For this topic, preferably find text books, as the proofs there are usually more carefully composed.