Approximation of operators in the topology of pointwise convergence (Q1901871)

This study was prompted by a question posed several years ago by Yu. A. Brudnyi: Is it true that the unit ball of \(L(X, Y)\) is dense in the unit ball of the space \(L(Y^*, X^*)\) in the topology of pointwise \(Y^* \times X\)-convergence (where \(L(X, Y)\) is the space of all linear continuous operators from \(X\) into \(Y\))? The elementary sufficient conditions for an affirmative answer to this question are the following: (a) space \(Y\) is reflexive; (b) \(X\) has the metric approximation property. As we will see, it is generally impossible to eliminate condition (b) even if we have a basis in \(Y\). Condition (a) can certainly be weakened.