Martínez Ansemil, José MaríaDineen, Seán2023-06-202023-06-2019871123-2536https://hdl.handle.net/20.500.14352/58693A subset L of a complex locally convex space E is said to be locally determining at 0 for holomorphic functions if for every connected open 0-neighborhood U and every f∈H(U), whenever f vanishes on U∩L, then f≡0. The authors' main result is that if E is separable and metrizable, then every set which is locally determining at 0 contains a null sequence which is also locally determining at 0. This answers a question of J. Chmielowski [Studia Math. 57 (1976), no. 2, 141–146;], who was the first to study locally determining sets. The proof of the main theorem makes use of the following result of K. F. Ng [Math. Scand. 29 (1971), 279–280;]: Let E be a normed space with closed unit ball BE. Suppose that there is a Hausdorff locally convex topology τ on E such that (BE,τ) is compact. Then E with its original norm is the dual of the normed space F={φ∈E∗: φ|BE is τ-continuous}, with norm ∥φ∥=sup{|φ(x)|: x∈BE}engjournal articlehttp://siba-ese.unisalento.it/index.php/notemat/article/view/1591/1371http://siba-ese.unisalento.it/index.php/index/indexopen access517.553Identity theoremLocally determining at zero for holomorphic functionsSeparable metrizable locally convex spaceNull sequenceEcuaciones diferenciales1202.07 Ecuaciones en Diferencias