Given a Banach space X and a subspace Y, the pair (X,Y) is said to have the approximation property (AP) provided there is a net of finite rank bounded linear operators on X all of which leave the subspace Y invariant such that the net converges uniformly on compact subsets of X to the identity operator. In particular, if the pair (X,Y) has the AP then X, Y, and the quotient space X/Y have the classical Grothendieck AP. The main result is an easy to apply dual formulation of this property. Applications are given to three-space properties; in particular, if X has the approximation property and its subspace Y is , then X/Y has the approximation property.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm8367-2-2016,
author = {T. Figiel and W. B. Johnson},
title = {The dual form of the approximation property for a Banach space and a subspace},
journal = {Studia Mathematica},
volume = {231},
year = {2015},
pages = {287-292},
zbl = {06575018},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm8367-2-2016}
}
T. Figiel; W. B. Johnson. The dual form of the approximation property for a Banach space and a subspace. Studia Mathematica, Tome 231 (2015) pp. 287-292. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm8367-2-2016/