New criterion of the approximation property
For Banach spaces \(X\) and \(Y\), let the Banach space of all bounded linear operators from \(X\) to \(Y\) be denoted by \({\mathcal L}(X,Y)\). Let \({\mathcal F}(X,Y)\) and \({\mathcal K}(X,Y)\) denote its subspaces of finite-rank and compact operators. A Banach space \(X\) is said to have the approximation property (AP) if the identity map on \(X\) belongs to the closure of \({\mathcal F}(X,X)\) in the topology \(\tau\) of uniform convergence on compact subsets of \(X\). By a well-known result of Grothendieck, \(X\) has the AP if and only if \({\mathcal F}(Y,X)\) is \(\| \cdot\| \)-dense in \({\mathcal K}(Y,X)\) for every Banach space \(Y\). The author proves that \(X\) has the AP if and only if \({\mathcal F}(X,Y)\) is \(\tau\)-dense in \({\mathcal K}(X,Y)\) for every Banach space \(Y\) (or just for every separable reflexive Banach space \(Y\)). Reviewer's remark. This gives a new proof to Grothendieck's result that the AP of \(X^\ast\) implies the AP of \(X\). Indeed, if \(X^\ast\) has AP, then \({\mathcal F}(X,Y)\) is \(\| \cdot\| \)-dense in \({\mathcal K}(X,Y)\) for all \(Y\) and hence also \(\tau\)-dense, which means that \(X\) has the AP. The traditional proof relies on the trace condition (``condition de biunivocité'') of Grothendieck.
scientific article; zbMATH DE number 5299492