We have, from real analysis, that: Let $a_n$ be an array in $\mathbb{R}$, such that $\sum_{n=0}^{+\infty} |a_n|$ converges, then, for any bijection $\phi:\mathbb{N} \to \mathbb{N}$ it holds $\sum_{n=1}^{+\infty} a_{\phi(n)} = \sum_{n=1}^{+\infty} a_n$(we call this part of Riemann theorem, the other part states that if $\sum_{n=1}^{+\infty} |a_n| does not converge, you can obtain any real number by permuting).
Motivated by this, I do not see why it wouldn't hold a generalisation of this:
Let $(X, ||\cdot||)$ be a Banach space, and $a_n$ such an array that $\sum_{n=1}^{+\infty} ||a_n||$ is finite, then for any bijection $\phi:\mathbb{N} \to \mathbb{N}$ it holds $\sum_{n=1}^{+\infty} a_n = \sum_{n=1}^{+\infty} a_{\phi(n)}$.
Does this really hold? Thank you!