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Locally determining sequences in infinite-dimensional spaces.

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Publication Date
1987
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Università del Salento
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A 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}
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J.CHMIELOWSKI: "Ensemble déterminants pour les fonctions analytiques", C.R.A.Sc.,Paris, t279.Serie A( 1974),63 9-641. J. CHMIELOWSKI: "Théoremes d'identité en dimension infinite", Ann.Poi.Math.,33(1976) . 35-37. J.CHMIELOWSKI: "Constructions des ensembles déterminants pour les fonctions analytiques",Studia Math.,LVII(1976),141-146. J.CHMIELOWSKI and G.LUBCZONOK:"A property of determining sets for Analytic Functions",Studia Math.,LX( 1977),285-288. S.DINEEN: "Complex analysis on locally convex spaces". North Holland Math. Studies, 57(1981). J.DIXMIER: "Sur un théoréme de Banaeh",Duke Math.J.,15(1948),1057-71. J.MUJICA:"A completeness criterion for inductive limits of Banach spaces",in Functional Analysis,Holomorphy and Approximation Theory II,Ed.G.Zapata,North Holland Math. Studies(1984), 319-329. K - F Ng: "On a theorem of Dixmier", Math. Scar.d., 29 ( 19 71)279-280.
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