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

### Volume 231 / 2015

Studia Mathematica 231 (2015), 287-292
MSC: Primary 46B03, 46B15, 46B20.
DOI: 10.4064/sm8367-2-2016
Published online: 3 March 2016

#### 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 ${\mathcal {L}}_\infty $, then $X/Y$ has the approximation property.