A generalized principle of local reflexivity (Q1312927)

The authors establish the following result: let \(E\) be a finite- dimensional subspace of \(X^{**}\) and let \(F\) be a reflexive subspace of \(X^*\). Then for every \(\varepsilon> 0\) there exists an injective linear operator \(S: E\to X\) such that \(\| S\| \| S^{-1}\|< 1+ \varepsilon\), \(Sx= x\) for all \(x\in E\cap X\), \(f(Sx^{**})= x^{**}(f)\) for any \(x^{**}\in E\), \(f\in F\). The proof is of tensorial type and depends upon the principle of local reflexivity [see \textit{A. Wilansky}, Port. Math. 38, 139-140 (1979; Zbl 0494.46016)]. Reviewer's remark: If \(F\) is finite-dimensional the result has been obtained by \textit{J. Lindenstrauss} and \textit{H. Rosenthal} [Isr. J. Math. 7, 325-349 (1969; Zbl 0205.126)] and \textit{W. B. Johnson}, \textit{H. Rosenthal} and \textit{M. Zippin} [ibid. 9, 488-506 (1971; Zbl 0217.161)]. For this case a quite elementary proof was presented at the Functional Analysis Conference at Oberwolfach, 1974, by \textit{C. Stegall} [see Proc. Am. Math. Soc. 78, 154-156 (1980; Zbl 0435.46020)].