Generalized Dini theorems for nets of functions on arbitrary sets

摘要: We characterize the uniform convergence of pointwise monotonic nets bounded real functions defined on arbitrary sets, without any particular structure. The resulting condition trivially holds for classical Dini theorem. Our vector-valued Dini-type theorem characterizes with relatively compact range in Hausdorff topological ordered vector spaces. As a consequence, such continuous space, we get equivalence between and convergence. When codomain is locally convex, also weak-pointwise convergence; this merges Dini-Weston from convex Most our results are free structural requirements common domain put compactness right place: functions.