On best approximations in Banach spaces from the perspective of orthogonality

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.