Summable vs absolutely summable

Question

I know there is a definition of "sumable family" in topological vector spaces as follows: the family of vectors $(v_\alpha)_{\alpha \in \Gamma}$ possesses for sum the vector $v$ if for every neighbourhood $\mathrm{N}$ of zero there exists a finite subset $\mathrm{F} \subset \Gamma$ such that $v - \sum\limits_{\alpha \in \mathrm{K}} v_\alpha \in \mathrm{N},$ regardless what finite set $\mathrm{K} \subset \Gamma$ was as long as $\mathrm{F}$ belonged (as a subset) to $\mathrm{K}.$ When dealing with normed spaces, this condition is weaker than absolute convergence for the case $\Gamma = \Bbb N.$ Is there a condition of "equal strenght" or stronger than absolute convergence?