(LB)-spaces and quasi-reflexivity

Abstract

Let $(X_n)$ be a sequence of infinite-dimensional Banach spaces. For $E$ being the space $\bigoplus_{n=1}^\infty X_n$, the following equivalences are shown: 1. Every closed subspace $Y$ of $E$, with the Mackey topology $\mu(Y,Y')$, is an (LB)-space. 2. Every separated quotient of $E'\ [\mu(E',E)]$\ is locally complete. 3. $X_n$ is quasi-reflexive,\ $n\in \mathbb{N}$. Besides this, the following two properties are seen to be equivalent: 1. $E'\ [\mu(E',E)]$ has the Krein-$\stackrel{\vee}{S}$mulian property. 2. $X_n$ is reflexive, $n\in \mathbb{N}$.