Orthogonality and Geometry in Normed Linear Spaces
Summary
Normed linear spaces provide the foundational setting for modern functional analysis by equipping vector spaces with a notion of length. Within these spaces, orthogonality extends the Euclidean concept of perpendicularity to more general settings, giving rise to several non‐equivalent definitions such as Birkhoff–James orthogonality, isosceles orthogonality and variants based on semi‐inner products. Geometry in this context encompasses the study of convexity, smoothness and rotundity of the unit ball, the behaviour of linear operators under the norm, and geometric constants that measure departure from inner-product structure. Key developments in recent years have clarified the relationships between different orthogonality notions, refined characterisations of strict convexity and smoothness via operator‐norm attainment, and advanced our understanding of geometric constants such as the von Neumann–Jordan and James constants. These insights not only deepen theoretical knowledge but also inform applications in optimisation theory, signal processing and quantum information, where the geometry of normed spaces governs stability, convergence and error estimates.
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Orthogonality and Geometry in Normed Linear Spaces publication trend
The graph below shows the total number of articles in orthogonality and geometry in normed linear spaces across all publications each year (not limited to Nature Index journals).
Technical terms
Normed linear space: A vector space endowed with a function (the norm) that assigns lengths to vectors and induces a metric.
Banach space: A complete normed linear space in which every Cauchy sequence converges.
Birkhoff–James orthogonality: A vector x is orthogonal to y if the norm of x does not decrease under any scalar perturbation in the direction of y.
Chmielinski orthogonality: An approximate form of Birkhoff–James orthogonality defined via duality mappings and geometric characterisations in Banach spaces.
Strict convexity: A property of a normed space whereby every point of the unit sphere is an extreme point of the unit ball, ensuring uniqueness of best approximations.
Smoothness: The differentiability of the norm at nonzero points, equivalent to the uniqueness of supporting hyperplanes at each boundary point of the unit ball.
References
- Properties of Chmielinski-orthogonality using Kadets-Klee property. Iraqi Journal for Computer Science and Mathematics (2022).
- On strong orthogonality and strictly convex normed linear spaces. Journal of Inequalities and Applications (2013).
- A pythagorean approach in Banach spaces. Journal of Inequalities and Applications (2006).
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