A general Orlicz-Pettis theorem (Q1860931)

Let \([X,Y]\) be a dual pair of topological vector spaces and \(\sigma (X,Y)\), \(\tau(X,Y)\) and \(\beta(X,Y)\) the weak topology, Mackey topology and the strong topology of \(X\), respectively, and let \(\lambda\) be a scalar-valued sequence space. A series \(\sum_ix_i\) is said to be \(\lambda\)-multiplier --\(\sigma (X,Y)\) convergent, if for each \((t_i)\in\lambda\), there exists an \(x\in X\), such that for each \(y\in Y\) there is \([x,y]= \sum^\infty_{i=1} [t_ix_i,y]\). In the paper it is shown that \(\lambda\)-multiplier convergence of series depends completely upon the \(AK\)-property of the sequence space \(\lambda\). From this result several important theorems are deduced.