A Topological Generalization of Orthogonality in Banach Spaces and some Applications

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.