The range of a projection of small norm in \(\ell^ n_ l\)
A Banach space X is called a \(P_{\lambda}\)-space if whenever X is contained in Y there is a projection P of Y onto X with \(\| P\|<\lambda\). It is not known, in general, what the \(P_{\lambda}\) spaces are for \(\lambda>1\). The conjecture that such k-dimensional spaces are close, using the Banach-Mazur distance, to \(\ell_{\infty}\) is equivalent by duality to showing that for any projection with norm \(<\lambda\) of \(\ell^ n_ 1\) onto a k-dimensional subspace, its range is close to \(\ell^ k_ l.\) The author addresses this question in the case \(\lambda\) is ''near'' 1 having proved the theorem: There exist a positive \(\epsilon_ 0\) and a positive function \(\phi\) (\(\epsilon)\) defined for \(0<\epsilon<\epsilon_ 0\) with \(\lim_{\epsilon \to 0}\phi(\epsilon)=0\) such that if k and n, \(n>k>0\), are integers and Q is a projection of \(\ell^ n_ l\) onto a k- dimensional subspace E with \(\| Q\| \leq 1+\epsilon\) then there is an integer \(h\geq k(1+\phi(\epsilon))\) and an h dimensional subspace F of E with \(d(F,\ell^ n_ l)\leq 1+\phi(\epsilon)\). Moreover, there is a projection P of \(\ell^ n_ l\) onto F with \(\| P\| \leq 1+\phi(\epsilon).\)
scientific article; zbMATH DE number 3867996