Norming subspaces of Banach spaces

TL;DR

This paper investigates conditions under which subspaces of Banach spaces and their duals are complemented, focusing on norming and totality properties, especially when certain subspaces are closed or reflexive.

Contribution

It establishes that under specific norming and totality conditions, subspaces and their pre-annihilators are complemented in the ambient Banach space, particularly when certain subspaces are closed or reflexive.

Findings

X and the pre-annihilator of Z are complemented in E under given conditions.

Complementation holds if Z is w*-closed or X is reflexive.

Provides conditions linking norming, totality, and complemented subspaces.

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 pre-annihilator of $Z$ are complemented in $E$ whenever $Z$ is $w^*$-closed or $X$ is reflexive.