The \(T\)-statistically convergent sequences are not an \(FK\)-space (Q1902936)

Let \(\omega\) denote the space of all real-valued sequences and \(T = (t_{nk})\) be a nonnegative regular matrix. For \(\varepsilon > 0\) and a scalar \(\ell\) let \(A_{\varepsilon \ell} = \{k : |x_k - \ell |< \varepsilon\}\). A sequence \(x = (x_k) \in \omega\) is defined to be \(T\)-statistically convergent to \(\ell\) if, for all \(\varepsilon > 0\), we have \(\lim_{n \to \infty} \sum t_{nk} \chi_{A \varepsilon \ell} (k) = 1\), where \(\chi_A\) denotes the characteristic function of \(A\). If \(T\) is the summation method \(C_1\) (the Cesàro matrix) the above definition coincides with the one of statistical convergence. The author states and proves the following result: For a nonnegative regular summation matrix \(T\) whose rows satisfy the condition \(\lim_{n \to \infty} \max_k t_{nk} = 0\), the space of \(T\)-statistically convergent sequences cannot be endowed with a locally convex \(FK\) topology.