Generalized sectional convergence and multipliers
A generalized sectional convergence scheme (gscs) is defined as a sequence \((T_n)\) of infinite matrices with the following properties: (1) Each \(T_n\) has finitely many nonzero entries, (2) for each \(i\), \(j\), \(\lim_{n\to\infty} T_n(i, j)=\delta_{ij}= \begin{cases} 1,(i=j)\\ 0,(i\neq j)\end{cases}\). Some of the results of this paper are: (1) If \(S\) is a barrelled \(AK(T_n)\) space, then \(S\) has the \(\beta\varphi\) topology. (2) Let \((T_n)\) be a gscs and let \(\Theta\) be a subspace of \(\ell^1\). Let \(\Lambda\) denote the largest subspace of \(\ell^1\) such that, setwise, \(\Lambda(T_n)= \Theta(T_n)\). The normed space \(\Lambda(T_n)\) is barrelled if and only if \(\Lambda\) is a barrelled subspace of \(\ell^1\). The paper also contains some multiplier theorems.
generalized sectional convergence scheme
scientific article; zbMATH DE number 811198
scientific article