A note on the Hyers-Ulam problem

Yunbai Dong. "A note on the Hyers-Ulam problem." Colloquium Mathematicae 138.2 (2015): 233-239. <http://eudml.org/doc/284324>.

@article{YunbaiDong2015,
abstract = {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.},
author = {Yunbai Dong},
journal = {Colloquium Mathematicae},
keywords = {isometry; Hyers-Ulam problem; Figiel's theorem},
language = {eng},
number = {2},
pages = {233-239},
title = {A note on the Hyers-Ulam problem},
url = {http://eudml.org/doc/284324},
volume = {138},
year = {2015},
}

TY - JOUR
AU - Yunbai Dong
TI - A note on the Hyers-Ulam problem
JO - Colloquium Mathematicae
PY - 2015
VL - 138
IS - 2
SP - 233
EP - 239
AB - 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.
LA - eng
KW - isometry; Hyers-Ulam problem; Figiel's theorem
UR - http://eudml.org/doc/284324
ER -