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Let ${\left(E_i\right)}_{i\in I}$ be a family of normed spaces and $\lambda$ a space of scalar generalized sequences. The $\lambda$-sum of the family ${\left(E_i\right)}_{i\in I}$ of spaces is $$\lambda \left\{{\left(E_i\right)}_{i\in I}\right\}:=\{{\left(x_i\right)}_{i\in I},x_i\in E_i,\text{and}(\parallel x_i{\parallel )}_{i\in I}\in \lambda \}.$$ Starting from the topology on $\lambda$ and the norm topology on each $E_i,$ a natural topology on $\lambda \left\{{\left(E_i\right)}_{i\in I}\right\}$ can be defined. We give conditions for $\lambda \left\{{\left(E_i\right)}_{i\in I}\right\}$ to be quasi-barrelled, barrelled or locally complete.