The Pure and Applied Mathematics (한국수학교육학회지시리즈B:순수및응용수학)

Korean Society of Mathematical Education (한국수학교육학회)
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CHARACTERIZATIONS OF AN INNER PRODUCT SPACE BY GRAPHS

  • Lin, C.S. (DEPARTMENT OF MATHEMATICS, BISHOP'S UNIVERSITY)
  • Published : 2009.11.30
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Abstract

The graph of the parallelogram law is well known, which gives rise to the characterization of an inner product space among normed linear spaces [6]. In this paper we will sketch graphs of its deformations according to our previous paper [7, Theorem 3.1 and 3.2]; each one of which characterizes an inner product space among normed linear spaces. Consequently, the graphs of some classical characterizations of an inner product space follow easily.

Keywords

References

  1. D. Amir: Characterizations of Inner Product Spaces. Birkhauser Verlag, Basel-Boston-Stuttgart, 1986.
  2. S.K. Berberian: Lectures in Functional Analysis and Operator Theory. Springer-Verlag, New York, 1974.
  3. S.O. Carlsson: A characterization property of Euclidean spaces. Arkiv fur Matem. 5 (1965), 327-330.
  4. M.M. Day: On criteria of Kasahara and Blumenthal for inner-product spaces. Proc. Amer. Math. Soc. 10 (1959), 92-100. https://doi.org/10.1090/S0002-9939-1959-0106407-0
  5. F.A. Ficken: Note on the existence of scalar products in normed linear spaces. Ann. Math. 45 (1944), 362-366. https://doi.org/10.2307/1969273
  6. P. Jordan & J.V. Neumann: On inner products in linear, metric spaces. Ann. Math. 36 (1935), 719-723. https://doi.org/10.2307/1968653
  7. C.-S. Lin: On strictly convex space, inner product space and their characterizations. Far east J. Math. Sci. 19 (2005), 121-132.
  8. C.-S. Lin: On (a,b,c,d)-orthogonality in normed linear spaces. Colloquium Math. 103 (2005), 1-10. https://doi.org/10.4064/cm103-1-1
  9. J. Oman: Norm identities which characterize inner product spaces. The Geometry of Metric and Linear Spaces. Michigan 1974, Springer Notes, no. 490, 116-133.