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
saindebmalya@gmail.com
Saikat Roy
Dept. of Mathematics, National Institute of Technology, Durgapur, West Bengal, India
saikatroy.cu@gmail.com
Kallol Paul
Dept. of Mathematics, Jadavpur University, Kolkata, West Bengal, India
kalloldada@gmail.com
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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