exp(A+B) = exp(A)exp(B) for matrices proof

Question

In this thread, On the proof: $\exp(A)\exp(B)=\exp(A+B)$ , where uses the hypothesis $AB=BA$?, it was mentioned that absolute convergence is required for swapping sums. What theorem is used precisely?

Answer 1

I think actually absolute convergence is only mentioned in the original question that is linked in the question you linked: https://math.stackexchange.com/a/356763/688699

Anyway if you're talking about rearranging an infinite sum then this is just the Riemann Series Theorem.

In particular (quoting from Wikipedia) let $X$ be a topological vector space. For example, this could be an additive matrix group, which we can see as $\mathbb{R}^{n}$ for some $n$. Then a series $\sum_{n=0}^{\infty} x_{n}$ is called unconditionally convergent if it converges to some point $x\in X$ and any rearrangement of the order of summation produces a series converging to $x$ also.

Then the Reimann Series Theorem says that, for $X=\mathbb{R}^{n}$, a series is unconditionally convergent if and only if it is absolutely convergent.

For more details see: https://en.wikipedia.org/wiki/Unconditional_convergence