\(M\)-ideals and the ``basic inequality'' (Q1314515)

It can be shown that, on a Banach space \(X\), every bounded linear operator has a best compact approximation, both by the basic inequality method and the method of \(M\)-ideals. In the present paper these methods are shown to be basically equivalent. The possibility of such a relationship was raised in \textit{S. Axler}, \textit{I. D. Berg}, \textit{N. Jewell} and \textit{A. Shields} [Ann. Math., II. Ser. 109, 601-612 (1979; Zbl 0399.47024)] and \textit{S. Axler}, \textit{N. Jewell} and \textit{A. Shields} [Trans. Am. Math. Soc. 261, 159-167 (1980; Zbl 0434.41016)].