1. Definition 2.10.1 - Local uniform convergence

Let $U \subseteq C$ be open, and $f_{n} : U \to C$ be a sequence of functions. We say that $f_{n}$ converges locally-uniformly to a function $f : U \to C$ if one of the following holds:

For any 1;;compact subset $K \subseteq U$ , $f_{n} \to f$ uniformly on 1;; $K$
For any $x \in U$ , there exists 1;; $B_{r} (x)$ , $r > 0$ such that $f_{n} \to f$ uniformly on 1;; $B_{r} (x)$

Remark

Actually, this is a weaker notion of uniform convergence. That is, uniformly convergent implies locally uniformally convergent.