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Some geometric characterizations of inear product spaces

Published online by Cambridge University Press:  17 April 2009

O. P. Kapoor  and
S. B. Mathur
O. P. Kapoor
Affiliation:
Department of Mathematics, Indian Institute of Technology, 11T Post Office, Kanpur - 208016, U.P., India.
S. B. Mathur
Affiliation:
Department of Mathematics, Indian Institute of Technology, 11T Post Office, Kanpur - 208016, U.P., India.
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Abstract

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There are several geometric characterizations of inner product spaces amongst the normed linear spaces. Mahlon M. Day's refinement “rhombi suffice as well as parallelograms”, of P. Jordan and J. von Neumann parallelogram law is well known. There are some characterizations which employ various notions of orthogonality. For example, it is known that if in a normed linear space Birkhoff-James orthogonality implies isosceles orthogonality. For example, it is known that if in a normed linear space Birkhoff-James orthogonality implies isosceles orthogonality then the space is an inner product space; geometrically it means that if the diagonals of a rectangle, with sides perpendicular in Birkhoff-James sense, are equal then the space is an inner product space. In the main result of this note we improve upon this characterization and show that here unit squares suffice as well as rectangles.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 24 , Issue 2 , October 1981 , pp. 239 - 246
Copyright
Copyright © Australian Mathematical Society 1981

References

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