On Exact and Approximate Orthogonalities Based on Norm Derivatives

  • Ali Zamani
  • Mahdi Dehghani


We survey mainly recent results on the orthogonality relations in normed linear spaces related to norm derivatives. We will focus on fundamental properties of norm derivatives orthogonality, differences and connections between these orthogonality types, and geometric results and problems closely related to them.


Norm derivative Orthogonality Approximate orthogonality Orthogonality preserving mappings Approximate orthogonality preserving property Stability 

Mathematics Subject Classification (2010)

Primary 46B20; Secondary 46C50 47B49 



The authors would like to thank the referee for her/his valuable suggestions and comments.


  1. 1.
    Abed, M.Y., Dehghani, M., Jahanipur, R.: Approximate \(\rho ^v_{\lambda }\)-orthogonality and its preservation. Ann. Funct. Anal.
  2. 2.
    Alonso, J., Martini, H., Wu, S.: On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces. Aequationes Math. 83, 153–189 (2012)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Alsina, C., Sikorska, J., Tomás, M.S.: Norm Derivatives and Characterizations of Inner Product Spaces. World Scientific, Hackensack (2010)zbMATHGoogle Scholar
  4. 4.
    Amir, D.: Characterizations of Inner Products Spaces. Birkauser, Basel (1986)Google Scholar
  5. 5.
    Ansari-piri, E., Sanati, R.G., Kardel, M.: A characterization of orthogonality preserving operators. Bull. Iran. Math. Soc. 43(7), 2495–2505 (2017)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Arambašić, L., Rajić, R.: Operators preserving the strong Birkhoff–James orthogonality on B(H). Linear Algebra Appl. 471, 394–404 (2015)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Balestro, V., Horváth, Á.G., Martini, H., Teixeira, R.: Angles in normed spaces. Aequationes Math. 91(2), 201–236 (2017)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Birkhoff, G.: Orthogonality in linear metric spaces. Duke Math. J. 1, 169–172 (1935)CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Blanco, A., Turnšek, A.: On maps that preserve orthogonality in normed spaces. Proc. R. Soc. Edinb. Sect. A 136(4), 709–716 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Brešar, M., Chebotar, M.A.: Linear preserver problems. In: Functional Identities, pp. 189–219. Birkhäuser, Basel (2007)Google Scholar
  11. 11.
    Chen, C., Lu, F.: Linear maps preserving orthogonality. Ann. Funct. Anal. 6(4), 70–76 (2015)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Chmieliński, J.: On an ε-Birkhoff orthogonality. J. Inequal. Pure Appl. Math. 6(3), Art. 79 (2005)Google Scholar
  13. 13.
    Chmieliński, J.: Linear mappings approximately preserving orthogonality. J. Math. Anal. Appl. 304, 158–169 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Chmieliński, J.: Stability of the orthogonality preserving property in finite-dimensional inner product spaces. J. Math. Anal. Appl. 318(2), 433–443 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Chmieliński, J.: Remarks on orthogonality preserving mappings in normed spaces and some stability problems. Banach J. Math. Anal. 1(1), 117–124 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Chmieliński, J.: Orthogonality preserving property and its Ulam stability. In: Rassias, T.M., Brzdȩk, J. (eds.) Functional Equations in Mathematical Analysis. Springer Optimization and Its Applications, vol. 52, pp. 33–58. Springer, Berlin (2011)Google Scholar
  17. 17.
    Chmieliński, J.: Normed spaces equivalent to inner product spaces and stability of functional equations. Aequationes Math. 87, 147–157 (2014)CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Chmieliński, J.: Orthogonality equation with two unknown functions. Aequationes Math. 90, 11–23 (2016)CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Chmieliński, J.: Operators reversing orthogonality in normed spaces. Adv. Oper. Theory 1(1), 8–14 (2016)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Chmieliński, J., Wójcik, P.: Isosceles-orthogonality preserving property and its stability. Nonlinear Anal. 72, 1445–1453 (2010)CrossRefMathSciNetzbMATHGoogle Scholar
  21. 21.
    Chmieliński, J., Wójcik, P.: On ρ-orthogonality. Aequationes Math. 80, 45–55 (2010)CrossRefMathSciNetzbMATHGoogle Scholar
  22. 22.
    Chmieliński, J., Wójcik, P.: ρ-orthogonality and its preservation–revisited. Recent developments in functional equations and inequalities, 17–30, Banach Center Publications, vol. 99, Polish Academy of Sciences, Institute of Mathematic, Warsaw (2013)Google Scholar
  23. 23.
    Chmieliński, J., Łukasik, R., Wójcik, P.: On the stability of the orthogonality equation and the orthogonality-preserving property with two unknown functions. Banach J. Math. Anal. 10(4), 828–847 (2016)CrossRefMathSciNetzbMATHGoogle Scholar
  24. 24.
    Dadipour, F., Sadeghi, F., Salemi, A.: Characterizations of inner product spaces involving homogeneity of isosceles orthogonality. Arch. Math. 104, 431–439 (2015)CrossRefMathSciNetzbMATHGoogle Scholar
  25. 25.
    Dadipour, F., Sadeghi, F., Salemi, A.: An orthogonality in normed linear spaces based on angular distance inequality. Aequationes Math. 90, 281–297 (2016)CrossRefMathSciNetzbMATHGoogle Scholar
  26. 26.
    Day, M.M.: Some characterizations of inner-product spaces. Trans. Am. Math. Soc. 62, 320–337 (1947)CrossRefMathSciNetzbMATHGoogle Scholar
  27. 27.
    Dehghani, M., Zamani, A.: Linear mappings approximately preserving ρ -orthogonality. Indag. Math. 28, 992–1001 (2017)CrossRefMathSciNetzbMATHGoogle Scholar
  28. 28.
    Dehghani, M., Abed, M., Jahanipur, R.: A generalized orthogonality relation via norm derivatives in real normed linear spaces. Aequationes Math. 93(4), 651–667 (2019)CrossRefMathSciNetzbMATHGoogle Scholar
  29. 29.
    Dragomir, S.S.: On approximation of continuous linear functionals in normed linear spaces. An. Univ. Timisoara Ser. Stiint. Mat. 29, 51–58 (1991)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Dragomir, S.S.: Continuous linear functionals and norm derivatives in real normed spaces. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 3, 5–12 (1992)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Dragomir, S.S.: Semi-Inner Products and Applications. Nova Science Publishers, Inc., Hauppauge (2004)zbMATHGoogle Scholar
  32. 32.
    Frank, M., Mishchenko, A.S., Pavlov, A.A.: Orthogonality-preserving, C -conformal and conformal module mappings on Hilbert C -modules. J. Funct. Anal. 260, 327–339 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  33. 33.
    Frank, M., Moslehian, M.S., Zamani, A.: Orthogonality preserving property for pairs of operators on Hilbert C -modules (2017). arXiv:1711.04724, Nov 10Google Scholar
  34. 34.
    Giles, J.R.: Classes of semi-inner product spaces. Trans. Am. Math. Soc. 116, 436–446 (1967)CrossRefMathSciNetzbMATHGoogle Scholar
  35. 35.
    Ilišević, D., Turnšek, A.: Approximately orthogonality preserving mappings on C -modules. J. Math. Anal. Appl. 341, 298–308 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  36. 36.
    Ionică, I.: On linear operators preserving orthogonality. Ann. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 58, 325–332 (2012)Google Scholar
  37. 37.
    Jahn, T.: Orthogonality in generalized Minkowski spaces. J. Convex Anal. 26(1), 49–76 (2019)MathSciNetzbMATHGoogle Scholar
  38. 38.
    James, R.C.: Orthogonality in normed linear spaces. Duke Math. J. 12(2), 291–302 (1945)CrossRefMathSciNetzbMATHGoogle Scholar
  39. 39.
    James, R.C.: Orthogonality and linear functionals in normed linear spaces. Trans. Am. Math. Soc. 61(2), 265–292 (1947)CrossRefMathSciNetGoogle Scholar
  40. 40.
    James, R.C.: Inner product in normed linear spaces. Bull. Am. Math. Soc. 53, 559–566 (1947)CrossRefMathSciNetzbMATHGoogle Scholar
  41. 41.
    Joichi, J.T.: Normed linear spaces equivalent to inner product spaces. Proc. Am. Math. Soc. 17, 423–426 (1966)CrossRefMathSciNetzbMATHGoogle Scholar
  42. 42.
    Jordan, P., von Neumann, J.: On inner products in linear, metric spaces. Ann. Math. 36(3), 719–723 (1935)CrossRefMathSciNetzbMATHGoogle Scholar
  43. 43.
    Karia, D.J., Parmar, Y.M.: Orthogonality preserving maps and pro-C -modules. J. Anal. 26, 1–10 (2018)CrossRefMathSciNetzbMATHGoogle Scholar
  44. 44.
    Kikianty, E., Sinnamon, G.: Angular equivalence of normed spaces. J. Math. Anal. Appl. 454(2), 942–960 (2017)CrossRefMathSciNetzbMATHGoogle Scholar
  45. 45.
    Koehler, D., Rosenthal, P.: On isometries of normed linear spaces. Stud. Math. 36, 213–216 (1970)CrossRefMathSciNetzbMATHGoogle Scholar
  46. 46.
    Koldobsky, A.: Operators preserving orthogonality are isometries. Proc. R. Soc. Edinb. Sect. A. 123(5), 835–837 (1993)CrossRefMathSciNetzbMATHGoogle Scholar
  47. 47.
    Kong, L., Cao, H.: Stability of orthogonality preserving mapping and the orthogonality equation. J. Shaanxi Normal Univ. Nat. Sci. Ed. 36(5), 10–14 (2008)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Köthe, G.: Topological Vector Spaces I. Springer, Berlin (1969)zbMATHGoogle Scholar
  49. 49.
    Leung, C.-W., Ng, C.-K., Wong, N.-C.: Linear orthogonality preservers of Hilbert bundles. J. Aust. Math. Soc. 89(2), 245–254 (2010)CrossRefMathSciNetzbMATHGoogle Scholar
  50. 50.
    Leung, C.-W., Ng, C.-K., Wong, N.-C.: Linear orthogonality preservers of Hilbert C -modules over C -algebras with real rank zero. Proc. Am. Math. Soc. 140(9), 3151–3160 (2012)CrossRefMathSciNetzbMATHGoogle Scholar
  51. 51.
    Leung, C.-W., Ng, C.-K., Wong, N.-C.: Linear orthogonality preservers of Hilbert C -modules. J. Oper. Theory 71(2), 571–584 (2014)CrossRefMathSciNetzbMATHGoogle Scholar
  52. 52.
    Li, C.K., Pierce, S.: Linear preserver problems. Am. Math. Mon. 108(7), 591–605 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  53. 53.
    Lumer, G.: Semi-inner product spaces. Trans. Am. Math. Soc. 100, 29–43 (1961)CrossRefMathSciNetzbMATHGoogle Scholar
  54. 54.
    Martini, H., Wu, S.: On maps preserving isosceles orthogonality in normed linear spaces. Note Mat. 29(1), 55–59 (2009)MathSciNetzbMATHGoogle Scholar
  55. 55.
    Miličić, P.M.: Sur le semi-produit scalaire dans quelques espaces vectorial norḿes. Mat. Vesn. 8(55), 181–185 (1971)zbMATHGoogle Scholar
  56. 56.
    Miličić, P.M.: Sur la G-orthogonalité dans les espéaceés normés. Mat. Vesn. 39, 325–334 (1987)zbMATHGoogle Scholar
  57. 57.
    Mojškerc, B., Turnšek, A.: Mappings approximately preserving orthogonality in normed spaces. Nonlinear Anal. 73, 3821–3831 (2010)CrossRefMathSciNetzbMATHGoogle Scholar
  58. 58.
    Moslehian, M.S.: On the stability of the orthogonal Pexiderized Cauchy equation. J. Math. Anal. Appl. 318, 211–223 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  59. 59.
    Moslehian, M.S., Zamani, A.: Mappings preserving approximate orthogonality in Hilbert C -modules. Math. Scand. 122, 257–276 (2018)CrossRefMathSciNetzbMATHGoogle Scholar
  60. 60.
    Moslehian, M.S., Zamani, A., Dehghani, M.: Characterizations of smooth spaces by ρ -orthogonality. Houst. J. Math. 43(4), 1187–1208 (2017)MathSciNetzbMATHGoogle Scholar
  61. 61.
    Moslehian, M.S., Zamani, A., Wójcik, P.: Approximately angle preserving mappings. Bull. Aust. Math. Soc. 99(3) 485–496 (2019)CrossRefMathSciNetzbMATHGoogle Scholar
  62. 62.
    Nabavi Sales, S.M.S.: On mappings which approximately preserve angles. Aequationes Math. 92, 1079–1090 (2018)CrossRefMathSciNetzbMATHGoogle Scholar
  63. 63.
    Pambuccian, V.: A logical look at characterizations of geometric transformations under mild hypotheses. Indag. Math. 11(3), 453–462 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  64. 64.
    Papini, P.L.: Uńasservatione sui prodotti semi-scalari negli spasi di Banach. Boll. Un. Mat. Ital. 6, 684–689 (1969)Google Scholar
  65. 65.
    Precupanu, T.: Duality mapping and Birkhoff orthogonality. An. Stiint, Univ. Al. I. Cuza Iasi. Mat. (S.N.) 59(1), 103–112 (2013)Google Scholar
  66. 66.
    Rätz, J.: On orthogonally additive mappings. Aequationes Math. 28, 35–49 (1985)CrossRefMathSciNetzbMATHGoogle Scholar
  67. 67.
    Roberts, B.D.: On the geometry of abstract vector spaces. Tôhoku Math. J. 39, 42–59 (1934)zbMATHGoogle Scholar
  68. 68.
    Sikorska, J.: Orthogonalities and functional equations. Aequationes Math. 89, 215–277 (2015)CrossRefMathSciNetzbMATHGoogle Scholar
  69. 69.
    Stypuła, T., Wójcik, P.: Characterizations of rotundity and smoothness by approximate orthogonalities. Ann. Math. Sil. 30, 193–201 (2016)MathSciNetzbMATHGoogle Scholar
  70. 70.
    Tapia, R.A.: A characterization of inner product spaces. Bull. Am. Math. Soc. 79, 530–531 (1973)CrossRefMathSciNetzbMATHGoogle Scholar
  71. 71.
    Tapia, R.A.: A characterization of inner product spaces. Proc. Am. Math. Soc. 41, 569–574 (1973)CrossRefzbMATHGoogle Scholar
  72. 72.
    Turnšek, A.: On mappings approximately preserving orthogonality. J. Math. Anal. Appl. 336, 625–631 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  73. 73.
    Ulam, S.M.: A Collection of Mathematical Problems. Interscience Publishers, New York (1960)zbMATHGoogle Scholar
  74. 74.
    Wang, S.-G., Ip, W.-C.: A matrix version of the Wielandt inequality and its applications to statistics. Linear Algebra Appl. 296, 171–181 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  75. 75.
    Wójcik, P.: Linear mappings preserving ρ-orthogonality. J. Math. Anal. Appl. 386, 171–176 (2012)CrossRefMathSciNetzbMATHGoogle Scholar
  76. 76.
    Wójcik, P.: Operators preserving and approximately preserving orthogonality and similar relations. Doctorial dissertation (in Polish), Katowice (2013)Google Scholar
  77. 77.
    Wójcik, P.: On mappings approximately transferring relations in finite-dimensional normed spaces. Linear Algebra Appl. 460, 125–135 (2014)CrossRefMathSciNetzbMATHGoogle Scholar
  78. 78.
    Wójcik, P.: Characterizations of smooth spaces by approximate orthogonalities. Aequationes Math. 89, 1189–1194 (2015)CrossRefMathSciNetzbMATHGoogle Scholar
  79. 79.
    Wójcik, P.: Operators preserving sesquilinear form. Linear Algebra Appl. 469, 531–538 (2015)CrossRefMathSciNetzbMATHGoogle Scholar
  80. 80.
    Wójcik, P.: Linear mappings approximately preserving orthogonality in real normed spaces. Banach J. Math. Anal. 9(2), 134–141 (2015)CrossRefMathSciNetzbMATHGoogle Scholar
  81. 81.
    Wójcik, P.: On certain basis connected with operator and its applications. J. Math. Anal. Appl. 423(2), 1320–1329 (2015)CrossRefMathSciNetzbMATHGoogle Scholar
  82. 82.
    Wójcik, P.: Operators reversing orthogonality and characterization of inner product spaces. Khayyam J. Math. 3(1), 23–25 (2017)MathSciNetzbMATHGoogle Scholar
  83. 83.
    Wójcik, P.: Mappings preserving B-orthogonality. Indag. Math. 30, 197–200 (2019)CrossRefMathSciNetzbMATHGoogle Scholar
  84. 84.
    Wu, S., He, C., Yang, G.: Orthogonalities, linear operators, and characterization of inner product spaces. Aequationes Math. 91(5), 969–978 (2017)CrossRefMathSciNetzbMATHGoogle Scholar
  85. 85.
    Zamani, A.: Approximately bisectrix-orthogonality preserving mappings. Comment. Math. 54(2), 167–176 (2014)MathSciNetzbMATHGoogle Scholar
  86. 86.
    Zamani, A., Moslehian, M.S.: Approximate Roberts orthogonality. Aequationes Math. 89, 529–541 (2015)CrossRefMathSciNetzbMATHGoogle Scholar
  87. 87.
    Zamani, A., Moslehian, M.S.: Approximate Roberts orthogonality sets and (δ, ε)-(a, b)-isosceles-orthogonality preserving mappings. Aequationes Math. 90, 647–659 (2016)CrossRefMathSciNetzbMATHGoogle Scholar
  88. 88.
    Zamani, A., Moslehian, M.S.: An extension of orthogonality relations based on norm derivatives. Q. J. Math. 70(2), 379–393 (2019)CrossRefMathSciNetzbMATHGoogle Scholar
  89. 89.
    Zamani, A., Moslehian, M.S., Frank, M.: Angle preserving mappings. Z. Anal. Anwend. 34, 485–500 (2015)CrossRefMathSciNetzbMATHGoogle Scholar
  90. 90.
    Zhang, Y.: An identity for (δ, ε)-approximately orthogonality preserving mappings. Linear Algebra Appl. 554, 358–370 (2018)CrossRefMathSciNetGoogle Scholar
  91. 91.
    Zhang, Y., Chen, Y., Hadwin, D., Kong, L.: AOP mappings and the distance to the scalar multiples of isometries. J. Math. Anal. Appl. 431(2), 1275–1284 (2015)CrossRefMathSciNetzbMATHGoogle Scholar

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Authors and Affiliations

  • Ali Zamani
    • 1
  • Mahdi Dehghani
    • 2
  1. 1.Department of MathematicsFarhangian UniversityTehranIran
  2. 2.Department of Pure MathematicsFaculty of Mathematical Sciences, University of KashanKashanIran

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