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(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}.

DOI Code: 10.1285/i15900932v31n1p191

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