2019/10/25 by Sain, Debmalya, Roy, Saikat, Paul, Kallol
#46A03 #FOS: Mathematics #Functional Analysis (math.FA) #Primary 57N17 #Secondary 47L05
paper · doi:10.48550/arxiv.1910.11751
We introduce the notion of orthogonality in a vector space with a topology on it. To serve our purpose, we define orthogonality space for a given vector space X, using the topology on it. 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.