What does the notation f|A mean?

[pellman]

Here is an instance of the notation in context.

If $$U_\lambda = F_\lambda(U_0)$$ and $$U_\lambda\cap U_0\neq\emptyset$$, then $$F_\lambda |U_{-\lambda}\cap U_0 :U_{-\lambda}\cap U_0 \rightarrow U_0 \cap U_\lambda$$ is a diffeomorphism and its inverse is $$F_{-\lambda}|U_0 \cap U_\lambda$$.

So what does the notation $$f|A$$ in $$f|A:A\rightarrow B$$ (where f is a function and A and B are sets) mean?

 

[Office_Shredder]

It's usually written f|_A as a subscript; it means f with its domain restricted to only A. While you don't have the original definition up, obviously the original domain of F in your statement was U₀ (or some superset thereof) and now they're looking at F restricted to the intersection with U_-lambda

 

[pellman]

That fits the other instances of its usage in the text. Thank you.