On best approximations in Banach spaces from the perspective of orthogonality

TL;DR

This paper investigates best approximation problems in Banach spaces using Birkhoff-James orthogonality, introducing algorithms, distance formulas, and inequalities that enhance classical results like Hölder's inequality.

Contribution

It presents new algorithms and formulas for best approximations in Banach spaces based on Birkhoff-James orthogonality, and strengthens classical inequalities with concrete examples.

Findings

Algorithms for best approximation in Banach spaces

New distance formulas based on orthogonality

Strengthened versions of Hölder's inequality

Abstract

We study best approximations in Banach spaces via Birkhoff-James orthogonality of functionals. To exhibit the usefulness of Birkhoff-James orthogonality techniques in the study of best approximation problems, some algorithms and distance formulae are presented. As an application of our study, we obtain some crucial inequalities, which also strengthen the classical H\"{o}lder's inequality. The relevance of the algorithms and the inequalities are discussed through concrete examples.