If a series has the same sum under any rearrangement, then is it absolutely convergent?

Question

Let $(V,\| \cdot \|)$ be a Banach space.

Let $\{a_n\}$ be a sequence in $V$ such that $\sum a_n$ converges.

Assume that for every bijection $f:\mathbb{N}\rightarrow \mathbb{N}, \sum a_n = \sum a_{f(n)}$.

In this case, does $\sum a_n$ have to be absolutely convergent?

When $V=\mathbb{R}$, $\sum a_n$ must be absolutely convergent, but in an arbitrary space, i'm not sure that this still holds

Answer 1

Take for example $V = \ell^2(\Bbb{N})$ and $a_n = 1/n \cdot \delta_n$ with $(\delta_n)_m =1$ for $n=m$ and $0$ otherwise.

Verification of the claimed properties is an exercise.