Summary: Let \((E,\tau)\) be a Hausdorff topological vector space with a basis \(\{e_i\}\), and \(\{f_i\}\) the sequence of coordinate functionals on \(E\) which is determined by \(\{e_i\}\). Let \(\sigma(E,F)\) be the weak topology on \(E\) which is generated by \(F= \{f_i\}\). In this paper, we show that each \(\sigma(E,F)\)-\({\mathcal K}\) convergent sequence in \(E\) is \(\tau\)-convergent to 0. As an application, we present a substantial improvement of the well-known Stiles’ Orlicz-Pettis theorem.