Why do we care for uniform convergence on compact sets?

Question

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.

Thanks in advance.

PS: this is my first question I ask here, so please don't be to hard on me, if it doesn't belong here.

Answer 1

Another reason could steam from Montel-type's results.

Assume you have a family of holomorphic functions that omit two values. Then the family will be normal.

Very intuitively: Being holomorphic functions rigid (imagine the rigidest ones, the constant functions), a family of them will "stick together in a somewhat clustered manner". This manner is having subsequences that converge uniformly on compacts.

Answer 2

I'd say being closed alone is sufficient reason. Usually the main reason for imbueing a topology onto a space is to be able to reason about convergence. Non-closed spaces are burdensome then, because there will be sequences which look like they converge (for metrizable spaces, think cauchy sequences), but don't because the limit element is "missing".