The dual form of the approximation property for a Banach space and a subspace

T. Figiel, and W. B. Johnson. "The dual form of the approximation property for a Banach space and a subspace." Studia Mathematica 231.3 (2015): 287-292. <http://eudml.org/doc/285460>.

@article{T2015,
abstract = {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.},
author = {T. Figiel, W. B. Johnson},
journal = {Studia Mathematica},
keywords = {approximation property for pairs; three-space property},
language = {eng},
number = {3},
pages = {287-292},
title = {The dual form of the approximation property for a Banach space and a subspace},
url = {http://eudml.org/doc/285460},
volume = {231},
year = {2015},
}

TY - JOUR
AU - T. Figiel
AU - W. B. Johnson
TI - The dual form of the approximation property for a Banach space and a subspace
JO - Studia Mathematica
PY - 2015
VL - 231
IS - 3
SP - 287
EP - 292
AB - 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.
LA - eng
KW - approximation property for pairs; three-space property
UR - http://eudml.org/doc/285460
ER -