# Why do we care for uniform convergence on compact sets?

I was trying to come up with reasons, why we naturally consider the topology of uniform convergence on compact sets as the appropriate framework for spaces of holomorphic functions such as e.g. $H(\mathbb{C}^n)$ (which is the space of entire functions on $\mathbb{C}^n$).

I understand that e.g. by Weierstrass' theorem the above space is closed, when $f_n$ converges compact to f. Moreover, that we can make such spaces into Fréchet spaces in case the domain behaves nicely enough (e.g. is the union of countable many compact sets, which clearly is the case for $\mathbb{C}^n$), which provides some nice topological properties.

But I would like to hear some more motivation or reasons, why to consider this topology naturally, which stem (purely) from complex analysis.