Asymptotically isometric copies of ℓ^{∞} in Banach spaces and a theorem of Bessaga and Pelczynski

Dowling, Patrick; Randrianantoanina, Narcisse

doi:10.1090/S0002-9939-00-05447-2

Asymptotically isometric copies of $\ell ^{\infty }$ in Banach spaces and a theorem of Bessaga and Pelczynski
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by Patrick N. Dowling and Narcisse Randrianantoanina PDF

Proc. Amer. Math. Soc. 128 (2000), 3391-3397 Request permission

Abstract:

We introduce the notion of a Banach space containing an asymptotically isometric copy of $\ell ^\infty$. A well known result of Bessaga and Pełczyński states a Banach space $X$ contains a complemented isomorphic copy of $\ell ^1$ if and only if $X^*$ contains an isomorphic copy of $c_0$ if and only if $X^*$ contains an isomorphic copy of $\ell ^\infty$. We prove an asymptotically isometric analogue of this result.

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