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A characterization of inner product spaces via norming vectors

Part of: Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  03 January 2025

Guillaume Aubrun  and
Mathis Cavichioli
Guillaume Aubrun*
Affiliation:
Université Lyon 1, CNRS, INRIA, Institut Camille Jordan, 43, boulevard du 11 novembre 1918, 69100 Villeurbanne, France
Mathis Cavichioli
Affiliation:
Département de mathématiques, Université Claude Bernard Lyon 1, 43, boulevard du 11 novembre 1918, 69100 Villeurbanne, France e-mail: [email protected]
*
e-mail: [email protected]
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Abstract

A finite-dimensional normed space is an inner product space if and only if the set of norming vectors of any endomorphism is a linear subspace. This theorem was proved by Sain and Paul for real scalars. In this paper, we give a different proof which also extends to the case of complex scalars.

Keywords

Characterization of inner-product spacesnorming vectorspositive John position

MSC classification

Primary: 46C15: Characterizations of Hilbert spaces
Type
Article
Information
Canadian Mathematical Bulletin , First View , pp. 1 - 5
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

The first author was supported in part by ANR under the grant ESQuisses (ANR-20-CE47-0014-01).

References

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Sain, D. and Paul, K., Operator norm attainment and inner product spaces . Linear Algebra Appl. 439(2013), no. 8, 2448–2452.CrossRefGoogle Scholar