Norm approximation property (Q2875919)

The author introduces the following variant of the approximation property, called norm approximation property: A Banach space \(X\) has the norm approximation property if there is a \(\lambda\geq1\) such that, for every \(\varepsilon>0\) and every finite-dimensional subspace \(E\subset X\), there is a bounded finite rank operator \(T: X\to X\) such that \(\|T\|\leq \lambda\) and for all \(x\in E\) NEWLINE\[NEWLINE (1-\varepsilon) \|x\| \leq \|Tx\| \leq (1+\varepsilon) \|x\|. NEWLINE\]NEWLINE The main results of the paper are: (1) There are Banach spaces without the norm approximation property (Pisier's space works). (2) Every closed subspace of \(\ell_p\) (\(1\leq p<\infty\)) or \(c_0\) has the norm approximation property; hence there are spaces with the norm approximation property that fail the approximation property. (3) Every Banach space is isometric to a \(1\)-complemented subspace of a space with the norm approximation property.