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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.
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 -