In 1958, Bessaga and Pełczyński proved the by now classical theorem:

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\noindent\textbf{(BP)} \textit{Let $X$ be a Banach space such that $X^*$ contains a subspace isomorphic to $c_0$. Then $X$ contains a complemented subspace isomorphic to $\ell_1$. Consequently, $X^*$ contains a subspace isomorphic to $\ell_\infty$}.

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Let $\Gamma$ be an infinite set. We introduce the notion of an asymptotically isometric (ai)-copy $Y_\Gamma$ of $\ell_1(\Gamma)$ in $X$, and an asymptotically orthogonal (ao)-projection from $X$ onto $Y_\Gamma$. The following four conditions are equaivalent:

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(1) $X^*$ contans an isometric copy of $c_0(\Gamma)$;

(2) $X$ contains an (ai)-copy $Y_\Gamma$ of $\ell_1(\Gamma)$ that is (ao)-complemented in $X^{**}$ (and hence in $X$);

(3) $\ell_1(\Gamma)$ is isometric to a quotient space of $X$;

(4) $X^*$ contains an isometric copy of $\ell_\infty(\Gamma)$.

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Combining condition (1) with the following result of
Pe{\l}czy\'nski we obtain generalizations of (BP) obtained earlier by Rosenthal, Dowling, Lewis, Randrianantoanina, and W\'ojtowicz:

\medskip \noindent\textit{If V and W are two Banach spaces and R is an isomorphic embedding of V into W, then there exists an equivalent norm on W under which R is an isometry}.