Orthogonality in smooth countably normed spaces

• Main Authors: Sarah Tawfeek, Nashat Faried, H. A. El-Sharkawy
• Format: Article
• Language: English
• Published: SpringerOpen 2021-01-01
• Series: Journal of Inequalities and Applications
• Subjects: Countably normed space; Normalized duality mapping; J-orthogonality; Uniformly convex countably normed space; Projection theorem in a countably normed space; Metric projection
• Online Access: https://doi.org/10.1186/s13660-020-02531-5

Abstract We generalize the concepts of normalized duality mapping, J-orthogonality and Birkhoff orthogonality from normed spaces to smooth countably normed spaces. We give some basic properties of J-orthogonality in smooth countably normed spaces and show a relation between J-orthogonality and metric projection on smooth uniformly convex complete countably normed spaces. Moreover, we define the J-dual cone and J-orthogonal complement on a nonempty subset S of a smooth countably normed space and prove some basic results about the J-dual cone and the J-orthogonal complement of S.