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

Casini, Emanuele; Papini, Pier Luigi

doi:10.1090/S0002-9939-1993-1152975-X

A counterexample to the infinity version of the Hyers and Ulam stability theorem
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by Emanuele Casini and Pier Luigi Papini PDF

Proc. Amer. Math. Soc. 118 (1993), 885-890 Request permission

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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Studies on the geometrical theory of Banach spaces

part

Almost convex sets in Banach spaces of type

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