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A note on the Hyers-Ulam problem

100%
Dong Y.
Colloquium Mathematicum
|
2015
|
tom 138
|
nr 2
233-239
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.
2
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Characterization of normed linear spaces with generalized Mazur intersection property

63%
Dong Y., Cheng Q.
Studia Mathematica
|
2013
|
tom 219
|
nr 3
193-200
EN
Let 𝓐 be a compatible collection of bounded subsets in a normed linear space. We give a characterization of the following generalized Mazur intersection property: every closed convex set A ∈ 𝓐 is an intersection of balls.
3
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Universal stability of Banach spaces for ε -isometries

51%
Cheng L., Dai D., Dong Y., Zhou Y.
Studia Mathematica
|
2014
|
tom 221
|
nr 2
141-149
EN
Let X, Y be real Banach spaces and ε > 0. A standard ε-isometry f: X → Y is said to be (α,γ)-stable (with respect to $T: L(f) ≡ \overline{span}f(X) → X$ for some α,γ > 0) if T is a linear operator with ||T|| ≤ α such that Tf- Id is uniformly bounded by γε on X. The pair (X,Y) is said to be stable if every standard ε-isometry f: X → Y is (α,γ)-stable for some α,γ > 0. The space X[Y] is said to be universally left [right]-stable if (X,Y) is always stable for every Y[X]. In this paper, we show that universally right-stable spaces are just Hilbert spaces; every injective space is universally left-stable; a Banach space X isomorphic to a subspace of $ℓ_{∞}$ is universally left-stable if and only if it is isomorphic to $ℓ_{∞}$; and a separable space X has the property that (X,Y) is left-stable for every separable Y if and only if X is isomorphic to c₀.
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