Dieudonné: Treatise on Analysis, chapter 16, differentiable manifolds: Common principle used for proofs

Question

In studying Dieudonné's Treatise on Analysis, vol. 3, chapter 16, differentialbe manifolds, I stumbled over a principle which he uses frequently for his proofs. I encounterd it for the first time in the proof of proposition 16.3.1, p.12, following the definition of a morphism of differentiable manifolds:

Here is a hand-drawn picture illustrating the situation for the proof of sufficiency of 16.3.1.

The point which I don't understand is why, at the end of the proof of sufficiency of condition 16.3.1, we can limit the discussion to the intersection $ U \cap U' $. It seems obvious to me that we can do this with respect to the chart of U, because the condition which holds for the chart of U also holds for the restricted chart of $ U \cap U' $. But then what we derive for $ U \cap U' $ as being the restricted chart of U', why does it hold for a chart on all of U'? I see no indication in all of the preceeding text that we are allowed to draw such a conclusion.

This principle of proving, limiting the discussion of what is valid for different charts to their intersection, thus always assuming that their domain be equal, appears again and again in the following, for example in 16.5.1.1. So it would be very helpful for me to properly understand this point.

PS: Of course I am aware that compatibility between charts is a question which is limited to the intersection of their domains. But this does not explain to me the above question.

Thanks for any help from someone maybe familiar with Dieudonné !

Answer 1

Differentiability is a purely local property, i.e. the question whether $f:X\rightarrow Y$ is differentiable at $x_0\in X$ can be decided on any neighbourhood $U $ of $x_0$.

So in the proof, you want to show differentiability in $U^\prime$. To this end you choose an arbitrary point $x_0$, and show differentiability in this point (by restricting to a smaller neighbourhood, if necessary, using the fact that differentiability is a local property). Since $x_0$ has been chosen arbitrarily, the conclusion will then hold in all of $U^\prime$.

(This is not a reasoning which is used only by Dieudonné)