Orthogonality of bilinear forms and application to matrices

TL;DR

This paper characterizes Birkhoff-James orthogonality for vector-valued functions and bilinear forms, providing a new elementary proof of the Bhatia-emrl Theorem in the real case, advancing understanding of orthogonality in functional analysis.

Contribution

It offers a new characterization of Birkhoff-James orthogonality for vector-valued functions and bilinear forms, with an elementary proof of a key theorem in the real case.

Findings

Characterization of Birkhoff-James orthogonality for continuous vector-valued functions.

Analysis of orthogonality of real bilinear forms.

Elementary proof of the Bhatia-emrl Theorem in the real setting.

Abstract

We characterize Birkhoff-James orthogonality of continuous vector-valued functions on a compact topological space. As an application of our investigation, Birkhoff-James orthogonality of real bilinear forms are studied. This allows us to present an elementary proof of the well-known Bhatia-\v{S}emrl Theorem in the real case.