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Journal of Convex Analysis 29 (2022), No. 3, 703--716
Copyright Heldermann Verlag 2022

A Topological Generalization of Orthogonality in Banach Spaces and some Applications

Debmalya Sain
Dept. of Mathematics, Indian Institute of Science, Bengaluru, Karnataka, India

Saikat Roy
Dept. of Mathematics, National Institute of Technology, Durgapur, West Bengal, India

Kallol Paul
Dept. of Mathematics, Jadavpur University, Kolkata, West Bengal, India

We introduce a topological notion of orthogonality in a vector space. We show that for a suitable choice of orthogonality space, Birkhoff-James orthogonality in a Banach space is a particular case of the orthogonality introduced by us. We characterize the right additivity of orthogonality in our setting and obtain a necessary and sufficient condition for a Banach space to be smooth, as a corollary to our characterization. Finally, using our notion of orthogonality, we obtain a topological generalization of the Bhatia-Semrl Theorem.

Keywords: Vector space with a topology, Birkhoff-James orthogonality, locally convex spaces, Bhatia-Semrl Theorem.

MSC: 57N17; 47L05, 46A03.

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