This is an announcement for the paper "A conjecture on the characterisation of inner product spaces" by Kallol Paul, Debmalya Sain and Lokenath Debnath.

Abstract: We study the properties of strongly orthonormal Hamel basis in the sense of Birkhoff-James in a finite dimensional real normed linear space that are analogous to the properties of orthonormal basis in an inner product space. We relate the notion of strongly orthonormal Hamel basis in the sense of Birkhoff-James with the notions of best approximation and best coapproximation in a finite dimensional real normed linear space. We prove that the existence of best coapproximation to any element of the normed linear space out of any one dimensional subspace and its coincidence with the best approximation to that element out of that subspace characterises a real inner product space of dimension( > 2). Finally we conjecture that a finite dimensional real smooth normed space of dimension ($>2$) is an inner product space iff given any element on the unit sphere there exists a strongly orthonormal Hamel basis in the sense of Birkhoff-James containing that element.

Archive classification: math.FA

Mathematics Subject Classification: Primary: 46B20, Secondary: 47A30

Submitted from: kalloldada@gmail.com

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1407.5016

or

http://arXiv.org/abs/1407.5016