Approximate symmetry of Birkhoff orthogonality. (English) Zbl 1402.46009
The present article explores two different notions of approximate Birkhoff-James orthogonality in a normed linear space and improves some of the previously known interrelations between them. It is illustrated that there are interesting connections with the geometric properties of the normed linear space, including smoothness, strict convexity and uniform convexity. The authors introduce the notion of approximate symmetry of Birkhoff-James orthogonality and relate it with the present study. Geometric properties of the normed linear space, related to the approximate symmetry of Birkhoff-James orthogonality, are also explored in the present paper, with special emphasis on uniformly convex Banach spaces and their dual spaces. Motivated by the results obtained in these directions, the authors introduce the Birkhoff-James symmetry constant for a given Banach space that provides a quantitative measurement of how far Birkhoff-James orthogonality in the space is from being symmetric. The concepts introduced in the present paper are novel and geometrically motivated.
Reviewer: Debmalya Sain (Contai)
MSC:
| 46B20 | Geometry and structure of normed linear spaces |
Keywords:
Birkhoff orthogonality; approximate Birkhoff orthogonality; approximate symmetry of Birkhoff orthogonalityReferences:
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