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Approximate norm‐additive maps on Banach spaces.
- Source :
-
Bulletin of the London Mathematical Society . Jul2024, p1. 20p. - Publication Year :
- 2024
-
Abstract
- Let X,Y$X, Y$ be two Banach spaces, F:X→Y$F:X\rightarrow Y$ be an ε$\varepsilon$‐norm‐additive map for some ε⩾0$\varepsilon \geqslant 0$, that is, |∥F(x)+F(y)∥−∥x+y∥|⩽ε,for allx,y∈X.$$\begin{equation*} \big\vert \Vert F(x)+F(y)\Vert -\Vert x+y\Vert \big\vert \leqslant \varepsilon, {\text{\,for all\,}} x,y\in X. \end{equation*}$$In this paper, we prove that if F$F$ is surjective with F(0)=0$F(0)=0$, then there exists a linear surjective isometry U:X→Y$U:X\rightarrow Y$ such that ∥F(x)−U(x)∥⩽32ε,for allx∈X.$$\begin{equation*} \Vert F(x)-U(x)\Vert \leqslant \frac{3}{2}\varepsilon, {\text{\,for all\,}} x\in X. \end{equation*}$$The estimate 32ε$\frac{3}{2}\varepsilon$ is sharp. We also approximate standard surjective ε$\varepsilon$‐norm‐additive maps between the positive cones of continuous function spaces by linear isometries within a sharp approximation error 32ε$\frac{3}{2}\varepsilon$. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00246093
- Database :
- Academic Search Index
- Journal :
- Bulletin of the London Mathematical Society
- Publication Type :
- Academic Journal
- Accession number :
- 178266968
- Full Text :
- https://doi.org/10.1112/blms.13115