The Problem
Let $(a_{n,m})_{n,m\in \mathbb{N}}$ be an sequence of complex numbers.
Under which conditions can I interchange product and limit?
$\lim_{m\to\infty}\prod_{n=1}^{\infty} a_{n,m}\overset{??}{=}\prod_{n=1}^{\infty} \lim_{m\to\infty} a_{n,m}$
My Idea: Uniform Convergence
Define $A_{N,m}:= \prod_{n=1}^Na_{n,m} $. Then we have
$ \lim_{m\to\infty}\prod_{n=1}^{\infty} a_{n,m}=\lim_{m\to\infty}\lim_{N\to\infty}\prod_{n=1}^{N} a_{n,m}=\lim_{m\to\infty}\lim_{N\to\infty}A_{N,m}$
If $a_{N,m}$ converges uniformly for either $N\to\infty$ or $m\to\infty$ , then we have
$ \lim_{m\to\infty}\lim_{N\to\infty}A_{N,m}=\lim_{N\to\infty}\lim_{m\to\infty}A_{N,m}=\lim_{N\to\infty}\prod_{n=1}^{N} \lim_{m\to\infty} a_{n,m}= \prod_{n=1}^{\infty} \lim_{m\to\infty} a_{n,m}$
The Question
Is there any other way?