Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On characterizations of inner product spaces


Author: Detlef Laugwitz
Journal: Proc. Amer. Math. Soc. 50 (1975), 184-188
MSC: Primary 46B05
MathSciNet review: 0374873
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The characterizations of inner product norms given by Tapia [10] make use of a generalized inner product in normed spaces. I shall give simpler proofs and a sharper result using a more geometrical approach.


References [Enhancements On Off] (What's this?)

  • [1] Wilhelm Blaschke, Kreis und Kugel, Walter de Gruyter & Co., Berlin, 1956 (German). 2te Aufl. MR 0077958
  • [2] T. Bonnesen and W. Fenchel, Theorie der konvexen Körper, Springer-Verlag, Berlin-New York, 1974 (German). Berichtigter Reprint. MR 0344997
  • [3] Ludwig Danzer, Detlef Laugwitz, and Hanfried Lenz, Über das Löwnersche Ellipsoid und sein Analogon unter den einem Eikörper einbeschriebenen Ellipsoiden, Arch. Math. (Basel) 8 (1957), 214–219 (German). MR 0094771
  • [4] Detlef Laugwitz, Differentialgeometrie in Vektorräumen, unter besonderer Berücksichtigung der unendlichdimensionalen Räume, Friedr. Vieweg & Sohn, Braunschweig, 1965 (German). MR 0182924
  • [5] Detlef Laugwitz, Konvexe Mittelpunktsbereiche und normierte Räume, Math. Z. 61 (1954), 235–244 (German). MR 0066667
  • [6] Detlef Laugwitz and Edgar R. Lorch, Riemann metrics associated with convex bodies and normed spaces, Amer. J. Math. 78 (1956), 889–894. MR 0082081
  • [7] E. R. Lorch, A curvature study of convex bodies in Banach spaces, Ann. Mat. Pura Appl. (4) 34 (1953), 105–112. MR 0052679
  • [8] J. Radon, Über eine besondere Art ebener konvexer Kurven, Leipziger Ber. Math.-Phys. Kl. 68 (1916), 123-128.
  • [9] Hanno Rund, The differential geometry of Finsler spaces, Die Grundlehren der Mathematischen Wissenschaften, Bd. 101, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1959. MR 0105726
  • [10] R. A. Tapia, A characterization of inner product spaces, Bull. Amer. Math. Soc. 79 (1973), 530-531; MR 47 #3964. Extended version, Proc. Amer. Math. Soc. 41 (1973), 569-574.
  • [11] Frederick A. Valentine, Konvexe Mengen, Übersetzung aus dem Englischen durch E. Heil. B. I.-Hochschultaschenbücher, Band 402/402a, Bibliographisches Institut, Mannheim, 1968 (German). MR 0226495

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46B05

Retrieve articles in all journals with MSC: 46B05


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0374873-0
Keywords: Inner product, differentiability of the norm, convex bodies
Article copyright: © Copyright 1975 American Mathematical Society