Let X,Y be real Banach spaces and ε > 0. Suppose that f:X → Y is a surjective map satisfying | ∥f(x)-f(y)∥ - ∥x-y∥ | ≤ ε for all x,y ∈ X. Hyers and Ulam asked whether there exists an isometry U and a constant K such that ∥f(x) - Ux∥ ≤ Kε for all x ∈ X. It is well-known that the answer to the Hyers-Ulam problem is positive and K = 2 is the best possible solution with assumption f(0) = U0 = 0. In this paper, using the idea of Figiel's theorem on nonsurjective isometries, we give a new proof of this result.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm138-2-7,
author = {Yunbai Dong},
title = {A note on the Hyers-Ulam problem},
journal = {Colloquium Mathematicae},
volume = {139},
year = {2015},
pages = {233-239},
zbl = {1327.46017},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm138-2-7}
}
Yunbai Dong. A note on the Hyers-Ulam problem. Colloquium Mathematicae, Tome 139 (2015) pp. 233-239. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm138-2-7/