Abstract
The present paper contributes to the ongoing programme of quantification of isomorphic Banach space theory focusing on the Hagler–Stegall characterisation of dual spaces containing complemented copies of L_{1}. As a corollary, we obtain the following quantitative version of the Hagler–Stegall theorem asserting that for a Banach space X, the following statements are equivalent:X contains almost isometric contains almost isometric copies of (bigoplus _{n=1}^{infty } ell _{infty }^{n})_{ell _1};for all varepsilon >0, X^{*} contains a (1+varepsilon )-complemented, (1+varepsilon )-isomorphic copy of L_{1};for all varepsilon >0, X^{*} contains a (1+varepsilon )-complemented, (1+varepsilon )-isomorphic copy of C[0,1]^{*}. Moreover, if X is separable, one may add the following assertion:for all varepsilon >0, there exists a (1+varepsilon )-quotient map T:Xrightarrow C(Delta ) so that T^{*}[C(Delta )^{*}] is (1+varepsilon )-complemented in X^{*}, where Delta is the Cantor set
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