On the entire bounded approximation property
Let X be a Banach space and \(X^*\) its dual space. A subspace \(M\subset X^*\) is said to be norming if there exists \(c>0\) such that \((\forall x\in X)(\sup_{0\neq f\in M}| f(x)| /\| f\| \geq c\| x\|).\) A subspace \(M\subset X^*\) is said to be quasibasic if there exists a sequence of linear continuous finite-dimensional operators \(v_ n: X\to X\), \(n\in {\mathbb{N}}\) such that: 1) (\(\forall x\in x)(\lim_{n\to \infty}\| v_ n(x)-x\| =0);\) 2) (\(\forall n\in {\mathbb{N}})(Im v^*_ n\subset M)\), where \(v^*_ n\)- operators, conjugate to \(v_ n.\) A Banach space X is said to have the entire bounded approximation property if every norming subspace \(M\subset X^*\) is quasibasic. This property is introduced in connection with investigations of regularizability of unbounded linear operators. The author proved that there exist Banach spaces isomorphic to their biduals which have bases but haven't entire bounded approximation property. This result complements previous results due to I. Singer, V. A. Vinokurov, A. N. Plichko, F. S. Vakher, L. V. Gladun.
regularizability of unbounded linear operators
scientific article; zbMATH DE number 4140521