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A generalization of the principle of local reflexivity. (English) Zbl 0609.46006

Our main result is the following generalization of the principle of local reflexivity:
For every Banach space X, finite-dimensional subspaces \(E\subset X''\), \(G\subset X'\), \(H\subset L(X)\), and \(\epsilon >0\) there is an isomorphism T from \(F:=E+lin\{R''E| R\in H\}\) into X such that
1. \(\| T\|\| T^{-1}\| \leq 1+\epsilon\)
2. \(T|_{E\cap X}=Id|_{E\cap X}\)
3. \(x'(Tx'')=x''(x')\) for x”\(\in F\), x’\(\in G\)
4. \(\| (TR''-RT)|_ E\| \leq \epsilon \| R\|\) for \(R\in H\).

MSC:

46B10 Duality and reflexivity in normed linear and Banach spaces
46B20 Geometry and structure of normed linear spaces
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