Communications in Mathematical Analysis

A Characterization of Inner Product Spaces Concerning an Euler-Lagrange Identity

Mohammad S. Moslehian and John M. Rassias

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Abstract

In this paper we present a new criterion on characterization of real inner product spaces concerning the Euler-Lagrange type identity

$$\|r_2x_1-r_1x_2\|^2 + \|r_1x_1+r_2x_2\|^2=(r_1^2+r_2^2)(\|x_1\|^2+\|x_2\|^2)\,.$$

Article information

Source
Commun. Math. Anal., Volume 8, Number 2 (2010), 16-21.

Dates
First available in Project Euclid: 21 April 2010

Permanent link to this document
https://projecteuclid.org/euclid.cma/1271890667

Mathematical Reviews number (MathSciNet)
MR2569953

Zentralblatt MATH identifier
1194.46037

Subjects
Primary: 46C015 46B20: Geometry and structure of normed linear spaces 46C05: Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)

Keywords
inner product space Day's condition normed space characterization of inner product spaces operator

Citation

Moslehian, Mohammad S.; Rassias, John M. A Characterization of Inner Product Spaces Concerning an Euler-Lagrange Identity. Commun. Math. Anal. 8 (2010), no. 2, 16--21. https://projecteuclid.org/euclid.cma/1271890667


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References

  • J. Alonso and A. Ullán, Moduli in normed linear spaces and characterization of inner product spaces. Arch. Math. (Basel) 59 (1992), no. 5, pp 487-495.
  • C. Alsina, P. Cruells and M. S. Tomás, Characterizations of inner product spaces through an isosceles trapezoid property. Arch. Math. (Brno) 35 (1999), no. 1, pp 21-27.
  • D. Amir, Characterizations of inner product spaces. Operator Theory: Advances and Applications 20. Birkhäuser Verlag, Basel, 1986.
  • E. Andalafte and R. Freese, Altitude properties and characterizations of inner product spaces. J. Geom. 69 (2000), no. 1-2, pp 1-10.
  • E. Z. Andalafte, C. R. Diminnie and R. W. Freese, $(\alpha,\beta)$-orthogonality and a characterization of inner product spaces. Math. Japon. 30 (1985), no. 3, pp 341-349.
  • S. Banach, Sur les opérations dans les ensembles abstraits et leur application auxéquations intégrales. Fund. Math. 3 (1922), pp 133-181.
  • M. Baronti and E. Casini, Characterizations of inner product spaces by orthogonal vectors. J. Funct. Spaces Appl. 4 (2006), no. 1, pp 1-6.
  • C. Benitez and M. del Rio, Characterization of inner product spaces through rectangle and square inequalities. Rev. Roumaine Math. Pures Appl. 29 (1984), no. 7, pp 543-546.
  • M. M. Day, Some characterizations of inner-product spaces. Trans. Amer. Math. Soc. 62 (1947), pp 320-337.
  • C. R. Diminnie, E. Z. Andalafte and R. Freese, Triangle congruence characterizations of inner product spaces. Math. Nachr. 144 (1989), pp 81-86.
  • S. S. Dragomir, Some characterizations of inner product spaces and applications. Studia Univ. Babes-Bolyai Math. 34 (1989), no. 1, 50-55.
  • P. Jordan and J. von Neumann, On inner products in linear, metric spaces. Ann. of Math. (2) 36 (1935), no. 3, pp 719-723.
  • J. Mendoza and T. Pakhrou, Characterizations of inner product spaces by means of norm one points. Math. Scand. 97 (2005), no. 1, pp 104-114.
  • M. S. Moslehian and J. M. Rassias, {Power and Euler-Lagrange norms. Aust. J. Math. Anal. Appl. 4 (2007), no. 1, Art. 17, 4 pp.
  • M. S. Moslehian and F. Zhang, An operator equality involving a continuous field of operators and its norm inequalities. Linear Algebra Appl. 429 (2008), no. 8-9, pp 2159-2167.
  • P. L. Papini, Inner products and norm derivatives. J. Math. Anal. Appl. 91 (1983), no. 2, 592-598.
  • [J. M. Rassias}, Two new criteria on characterizations of inner products. Discuss. Math. 9 (1988), pp 255-267 (1989).
  • J. M. Rassias, Four new criteria on characterizations of inner products. Discuss. Math. 10 (1990), pp 139-146 (1991).
  • Th. M. Rassias, New characterizations of inner product spaces. Bull. Sci. Math. (2) 108 (1984), no. 1, pp 95-99.
  • Th. M. Rassias, On characterizations of inner product spaces and generalizations of the H. Bohr inequality. Topics in mathematical analysis, 803-819, Ser. Pure Math., 11, World Sci. Publ., Teaneck, NJ, 1989.
  • J. Rätz, Characterization of inner product spaces by means of orthogonally additive mappings. Aequationes Math. 58 (1999), no. 1-2, pp 111-117.
  • I. J. Schoenberg, A remark on M. M. Day's characterization of inner-product spaces and a conjecture of L. M. Blumenthal. Proc. Amer. Math. Soc. 3, (1952). pp 961-964.
  • P. Šemrl, Additive functions and two characterizations of inner-product spaces. Glas. Mat., III. Ser. 25(45) (1990), no. 2, pp 309-317.
  • I. Şerb, Rectangular modulus, Birkhoff orthogonality and characterizations of inner product spaces. Comment. Math. Univ. Carolin. 40 (1999), no. 1, pp 107-119.