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

Volume 231 / 2015

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


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.


  • T. FigielInstitute of Mathematics
    Polish Academy of Sciences
    Branch in Gdańsk
    Wita Stwosza 57
    80-952 Gdańsk, Poland
  • W. B. JohnsonDepartment of Mathematics
    Texas A&M University
    College Station, TX 77843-3368, U.S.A.

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