A note on the Hyers-Ulam problem

• Autorzy: Yunbai Dong
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 46B20: Geometry and structure of normed linear spaces
• 46B04: Isometric theory of Banach spaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2015
• Tom: 138
• Numer: 2
• Strony: 233-239

Daty

wydano

2015

Twórcy

autor

Yunbai Dong

• School of Mathematics and Computer Science, Wuhan Textile University, Wuhan 430073, China

Identyfikatory

DOI

10.4064/cm138-2-7