  We study best approximations to compact operators between Banach spaces and
Hilbert spaces, from the point of view of Birkhoff-James orthogonality and
semi-inner-products. As an application of the present study, some distance
formulae are presented in the space of compact operators. The special case of
bounded linear functionals as compact operators is treated separately and some
applications to best approximations in reflexive, strictly convex and smooth
Banach spaces are discussed. An explicit example is presented in $ \ell_p^{n} $
spaces, where $ 1 < p < \infty, $ to illustrate the applicability of the
methods developed in this article. A comparative analysis of the results
presented in this article with the well-known classical duality principle in
approximation theory is conducted to demonstrate the advantage in the former
case, from a computational point of view.
