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Analytic Elements with a Null Derivative on an Infraconnected Open Set
Author(s) -
Escassut Alain,
Diarra Bertin
Publication year - 1990
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-42.1.137
Subject(s) - ultrametric space , open set , algebraically closed field , mathematics , constant (computer programming) , subalgebra , zero (linguistics) , set (abstract data type) , space (punctuation) , field (mathematics) , derivative (finance) , discrete mathematics , combinatorics , pure mathematics , algebra over a field , computer science , metric space , financial economics , economics , programming language , linguistics , philosophy , operating system
Let K be a complete ultrametric valued algebraically closed field of characteristic zero, let D be an open set in K and let H(D) be the topological linear space of the analytic elements in D . We first extend a previous result (already known for Banach algebras H(D) [8]) in proving when H(D) is a subalgebra of K D , D is infraconnected if and only if the assertion f ′ ‵ 0 implies that f = constant whenever f ∈ H(D) . Later we show that such a characterization of the infraconnected sets cannot be extended in a significant way, by constructing an open infraconnected set D together with a non‐constant element f ∈ H(D) such that f ′ ‵ 0.