An improvement of a recent closed graph theorem
Let \(X\) and \(Y\) be topological vector spaces. The authors call a mapping \(f:X\to Y\) weakly quasilinear if \(f\) satisfies: (1)~if \(f(x_n)\to 0\) and \(f(u_n)\to0\), then \(f(x_n+u_n)\to0\); (2)~if \(f(x_n-x)\to0\) and \(t_n\to t\) in the scalar field \(K\), then \(f(t_nx_n-tx)\to0\); and (3)~if \(x_n-u\to0\) in \(X\), then \(f(x_n)\to f(u)\) if and only if \(f(x_n-u)\to0\). They show that this property of \(f\) is properly weaker than the property of being quasi-linear in the sense of a paper of the first two authors [J. Math. Anal. Appl. 311, No.~2, 736--743 (2005; Zbl 1087.46003)]. Then, they extend the closed graph theorem from the mentioned paper to the case of weakly quasi-linear mappings.
scientific article; zbMATH DE number 5346273