Local completeness and dual local quasi-completeness

It is proved that $l^{q}$-completeness $(1<q<\infty)$ is equivalent to $l^{1}$-completeness (defined by Saxon and Sánchez Ruiz), and becomes a new characteristic condition for local completeness. The relationship between dual local completeness, dual local quasi-completeness and the Banach-Mackey property is investigated. For a quasi-Mackey space, dual local quasi-completeness, $c_{0}$-quasi-barrelledness, Ruess' property (quasi-L) and $C$-quasi-barrelledness are equivalent to each other.