Abstract

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.

Author information

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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

Keywords

  • Vector space with a topology
  • Birkhoff-James orthogonality
  • locally convex spaces
  • Bhatia-Semrl Theorem

Mathematics Subject Classification

57N17; 47L05, 46A03

References

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Suggested citation

D. Sain, S. Roy, K. Paul. “A Topological Generalization of Orthogonality in Banach Spaces and some Applications.” Journal of Convex Analysis 29 (2022), No. 3, 703–716. https://doi.org/10.68381/jca29039

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Published by Heldermann Verlag, 2022. Rights now held by Banach Press LLC.