Orthogonality in a vector space with a topology And a generalization of Bhatia-Semrl Theorem

TL;DR

This paper introduces a new concept of orthogonality in topological vector spaces, generalizes the Bhatia-Semrl Theorem, and characterizes smooth Banach spaces through this framework.

Contribution

It defines a novel orthogonality in topological vector spaces, unifies it with Birkhoff-James orthogonality, and provides a topological generalization of the Bhatia-Semrl Theorem.

Findings

Unified orthogonality framework in topological vector spaces.

Characterization of smooth Banach spaces via orthogonality.

Topological generalization of the Bhatia-Semrl Theorem.

Abstract

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.