In French, given a normed vector space $(E, \Vert \cdot \Vert)$, we say that a set of vectors $\mathcal C = \{c_i \ ; \ i \in I\}$ is a "famille sommable" when $$(\forall \epsilon > 0) \, (\exists J_0 \in \mathcal F(I)) \, (\forall K \in \mathcal F(I \setminus J_0)) \, \left\Vert \displaystyle \sum_{k \in K} c_k \right\Vert< \epsilon$$
where $\mathcal F(A)$ is defined as the sets of finite subsets of $A$.
Is there a similar wording in English? At least Wikipedia page I provided above is not refering to a similar an English page.
Note: this question has for root cause Uncountable sum of vectors in a Hilbert Space.
It's a “summable family”. That's the expression that is used in the English version of Bourbaki's General Topology.