Let us consider the topological vector space $\mathbb{C}$ equipped with the euclidean topology.
We know that a series of complex numbers $\sum_{n=1}^{\infty}a_n$ is said to be absolutely convergent iff $\sum_{n=1}^{\infty}||a_n||$ is convergent.
I've always assumed that by $||\cdot||$ we mean, when considering absolute convergence, the euclidean norm. Now that I'm reviewing my notes, I'm wondering why we usually make this assumption. What happens if I consider different kinds of norm?
In particular I'd like to know
- Does absolute convergence depend on the choice of $||\cdot||$ ?
- If I change the norm, does absolute convergence still imply convergence?
- Is there any book or paper addressing this question?