(LB)-spaces and quasi-reflexivity

doi:10.1285/i15900932v31n1p191

(LB)-spaces and quasi-reflexivity

Manuel Valdivia

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, $n\in \mathbb{N}$ .

DOI Code: 10.1285/i15900932v31n1p191

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(LB)-spaces and quasi-reflexivity

Abstract