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A counterexample to the infinity version of the Hyers and Ulam stability theorem

Authors: Emanuele Casini and Pier Luigi Papini
Journal: Proc. Amer. Math. Soc. 118 (1993), 885-890
MSC: Primary 26E15; Secondary 26B25, 46G99
MathSciNet review: 1152975
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Abstract: Hyers and Ulam proved a stability result for convex functions, defined in a subset of $ {\Re ^n}$. Here we give an example showing that their result cannot be extended to those functions defined in infinite-dimensional normed spaces. Also, we give a positive result for a particular class of approximately convex functions, defined in a Banach space, whose norm satisfies the so-called convex approximation property.

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Keywords: Approximately convex functions, Hyers and Ulam theorem, convex approximation property
Article copyright: © Copyright 1993 American Mathematical Society

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