Aequationes mathematicae

, Volume 89, Issue 3, pp 529–541 | Cite as

Approximate Roberts orthogonality



In a real normed space we introduce two notions of approximate Roberts orthogonality as follows:
$$x \perp_R^\varepsilon y \, {\rm if \, and \, only \, if} \left|\|x + ty\|^2 - \|x - ty\|^2\right| \leq 4\varepsilon\|x\|\|ty\| \, {\rm for \, all} \, t \in \mathbb{R}\,;$$
$$x^{\varepsilon} \perp_R y \, {\rm if \, and \, only \, if} \left|\|x + ty\|-\|x - ty\|\right| \leq \varepsilon(\|x + ty\| + \|x - ty\|) \, {\rm for \, all} \, t \in \mathbb{R}\,.$$
We investigate their properties and their relationship with the approximate Birkhoff orthogonality. Moreover, we study the class of linear mappings preserving approximately Roberts orthogonality of type  \({^{\varepsilon}\perp_R}\). A linear mapping \({U: \mathcal{X} \to \mathcal{Y}}\) between real normed spaces is called an \({\varepsilon}\)-isometry if \({(1 - \varphi_1 (\varepsilon))\|x\| \leq \|Ux\| \leq (1 + \varphi_2(\varepsilon))\|x\|\,\,(x \in \mathcal{X})}\), where \({\varphi_1 (\varepsilon)\rightarrow0}\) and \({\varphi_2 (\varepsilon)\rightarrow0}\) as \({\varepsilon\rightarrow 0}\). We show that a scalar multiple of an \({\varepsilon}\)-isometry is an approximately Roberts orthogonality preserving mapping.

Mathematics Subject Classification (1991)

Primary 46B20 Secondary 46C05 


Roberts orthogonality approximate orthogonality \({\varepsilon}\)-isometry orthogonality preserving mapping 


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  1. 1.
    Alonso J., Benítez C.: Carlos Orthogonality in normed linear spaces: a survey. II. Relations between main orthogonalities. Extracta Math. 4(3), 121–131 (1989)MathSciNetGoogle Scholar
  2. 2.
    Alonso J., Martini H., Wu S.: On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces. Aequationes Math. 83(1-2), 153–189 (2012)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Alsina C., Sikorska J., Tomás M.S.: Norm Derivatives and Characterizations of Inner Product Spaces. World Scientific, Hackensack (2009)CrossRefGoogle Scholar
  4. 4.
    Amir D.: Characterization of Inner Product Spaces. Birhäuser Verlag, Basel (1986)CrossRefGoogle Scholar
  5. 5.
    Birkhoff G.: Orthogonality in linear metric spaces. Duke Math. J. 1, 169–172 (1935)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Chmieliński, J.: On an \({\varepsilon}\)-Birkhoff orthogonality. J. Inequal. Pure Appl. Math. 6(3), Art. 79 (2005)Google Scholar
  7. 7.
    Chmieliński J.: Remarks on orthogonality pereserving mappings in normed spaces and some stability problems. Banach J. Math. Anal. 1(1), 117–124 (2007)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Chmieliński J.: Orthogonality preserving property and its Ulam stability, Chapter 4. In: Brzdek, J., Rassias Th., M. (eds.) Functional Equations in Mathematical Analysis. Springer Optimization and its Applications, vol. 52, pp. 33–58. Springer, New York (2012)Google Scholar
  9. 9.
    Chmieliński J., Wójcik P.: Isosceles-orthogonality preserving property and its stability. Nonlinear Anal. 72, 1445–1453 (2010)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Chmieliński J., Wójcik P.: On a ρ-orthogonality. Aequationes Math. 80, 45–55 (2010)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Dragomir S.S.: On approximation of continuous linear functionals in normed linear spaces. An. Univ. Timişoara Ser. Ştiinţ. Mat. 29, 51–58 (1991)MATHGoogle Scholar
  12. 12.
    James R.C.: Orthogonality in normed linear spaces. Duke Math. J. 12, 291–301 (1945)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Maligranda L.: Some remarks on the triangle inequality for norms. Banach J. Math. Anal. 2(2), 31–41 (2008)MATHMathSciNetGoogle Scholar
  14. 14.
    Miličić P.M.: Sur la G-orthogonalité dans les espéaceésnormés. Math. Vesnik. 39, 325–334 (1987)MATHGoogle Scholar
  15. 15.
    Mirzavaziri M., Moslehian M.S.: Orthogonal constant mappings in isoceles orthogonal spaces. Kragujevac J. Math. 29, 133–140 (2006)MATHMathSciNetGoogle Scholar
  16. 16.
    Mojškerc B., Turnšek A.: Mappings approximately preserving orthogonality in normed spaces. Nonlinear Anal. 73, 3821–3831 (2010)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Moslehian M.S.: On the stability of the orthogonal Pexiderized Cauchy equation. J. Math. Anal. Appl. 318(1), 211–223 (2006)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Odell E., Schlumprecht Th.: Asymptotic properties of Banach spaces under renormings. J. Am. Math. Soc. 11(11), 175–188 (1998)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Roberts B.D.: On the geometry of abstract vector spaces. Tôhoku Math. J. 39, 42–59 (1934)Google Scholar

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© Springer Basel 2013

Authors and Affiliations

  1. 1.Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures (CEAAS)Ferdowsi University of MashhadMashhadIran

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