On the topology of the space of entire functions.

Question

I have a doubt. When we have a Banach space $X$, by definition, each element $f$ in $X$ has a norm $\left\|f\right\|_X$. On the other hand, I understand that the space of entire functions $\mathcal{O}(\mathbb{C}):=H$ is not a Banach space but if it is a Frechet space and the topology obtained is the one obtained from the norms $\left\|f\right\|_K:=sup_{z\in K}|f(z)|$ with $K$ a compact subset of $\mathbb{C}$. Is it possible to define $\left\|f\right\|_H$ in some way? I mean if $\left\|f\right\|_H=sup_{z\in K}|f(z)|$ for some $K$ compact subset of $\mathbb{C}$? (I can't understand this concept) Thank you.