On best approximations to compact operators

We study best approximations to compact operators between Banach spaces and Hilbert spaces, from the point of view Birkhoff-James orthogonality semi-inner-products. As an application present study, some distance formulae are presented in space operators. The special case bounded linear functionals as is treated separately applications reflexive, strictly convex smooth discussed. An explicit example ? p n \ell _p^{n} where alttext="1 greater-than normal infinity comma"> 1 &gt; ?<!-- ? <mml:mo>, encoding="application/x-tex">1 &gt; \infty , illustrate applicability methods developed this article. A comparative analysis results article with well-known classical duality principle approximation theory conducted demonstrate advantage former case, a computational view.