  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.
