Let $f_n$ be a sequence of functions and $f$ be a function, all having the same domain and their range shall belong to the same metric space. What is the usual meaning of this notation:
$f_n\rightrightarrows f$?
I encountered this notation a couple of times now, in a context where it was supposed to mean uniform convergence I guess.
Also does anyone also know an author of a book that is using this notation?
This is a relatively uncommon notation meaning that $f_n$ converges uniformly to $f$. Most analysis books do not use any special notation for uniform convergence, instead preferring to write "$f_n\to f$ uniformly" or similar. However, it is used in Pugh's text (p. 212).
Charles C. Pugh (2015). Real Mathematical Analysis, Second Edition. Springer.