What does "$C^{\infty}$" convergence mean?

Question

I'm studying first notions about several complex variables.

As a consequence of the (generalized form) of the Cauchy esteem for holomorphic functions, the book says that in the space $\mathcal H(\Omega)$ (the space of holomorphic functions $f:\Omega\to\Bbb C$, where $\Omega\subseteq\Bbb C^n$ is open) $C^{\infty}$-convergence coincides with $L_{{loc}}^1$-convergence.

Can someone explain me what $C^{\infty}$ convergence is? I thought It could be the convergence in the usual norm which with $C^{\infty}$ is usually endowed, which is the sup-norm. Am I right?

Thanks to all.

Answer 1

To converge in $C^\infty$ every derivative has to converge uniformly on each compact set. So for all $K$ compact, and multi-index $\alpha$ we have

$$ \sup \limits_{z \in K} || D^\alpha (f_n - f) || \to 0 $$