We show that if $ X$ is a closed subspace of a Banach space $ E$ and $ Z$ is a closed subspace of $ E^*$ such that $ Z$ is norming for $ X$ and $ X$ is total over $ Z$ (as well as $ X$ is norming for $ Z$ and $ Z$ is total over $ X$), then $ X$ and the preannihilator of $ Z$ are complemented in $ E$ whenever $ Z$ is $ w^*$-closed or $ X$ is reflexive.
Norming subspaces of Banach spaces / V.P. Fonf, S. Lajara, S. Troyanski, C. Zanco. - In: PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY. - ISSN 1088-6826. - 147:7(2019 Mar 26), pp. 3039-3045. [10.1090/proc/14442]
Norming subspaces of Banach spaces
C. ZancoUltimo
2019
Abstract
We show that if $ X$ is a closed subspace of a Banach space $ E$ and $ Z$ is a closed subspace of $ E^*$ such that $ Z$ is norming for $ X$ and $ X$ is total over $ Z$ (as well as $ X$ is norming for $ Z$ and $ Z$ is total over $ X$), then $ X$ and the preannihilator of $ Z$ are complemented in $ E$ whenever $ Z$ is $ w^*$-closed or $ X$ is reflexive.
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