# Article

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Keywords:
isometry; embedding of $\ell_\infty$; dual space; Banach lattice
Summary:
Let $X$ and $E$ be a Banach space and a real Banach lattice, respectively, and let $\Gamma$ denote an infinite set. We give concise proofs of the following results: (1) The dual space $X^*$ contains an isometric copy of $c_0$ iff $X^*$ contains an isometric copy of $\ell_\infty$, and (2) $E^*$ contains a lattice-isometric copy of $c_0(\Gamma)$ iff $E^*$ contains a lattice-isometric copy of $\ell_\infty(\Gamma)$.
References:
[1] Abramovich Y.A., Wickstead A.W.: When each continuous operator is regular. II. Indag. Math., N.S. 8 (1997), 281-294. MR 1622216 | Zbl 0908.47031
[2] Aliprantis C.D., Burkinshaw O.: Positive Operators. Academic Press, New York, 1985. MR 0809372 | Zbl 1098.47001
[3] Bessaga C., Pełczyński A.: On basis and unconditional convergence of series in Banach spaces. Studia Math. 17 (1958), 151-164. MR 0115069
[4] Dowling P.N.: Isometric copies of $c_0$ and $\ell_\infty$ in duals of Banach spaces. J. Math. Anal. Appl. 244 (2000), 223-227. MR 1746799 | Zbl 0955.46011
[5] Lindenstrauss J., Tzafriri L.: Classical Banach Spaces. I. Springer, Berlin, 1977. MR 0500056 | Zbl 0362.46013
[6] Meyer-Nieberg P.: Banach Lattices. Springer, Berlin, 1991. MR 1128093 | Zbl 0743.46015
[7] Rosenthal H.P.: On injective Banach spaces and the spaces $L^\infty(\mu)$ for finite measures $\mu$. Acta Math. 124 (1974), 205-247. MR 0257721
[8] Rosenthal H.P.: On relatively disjoint families of measures, with some applications to Banach space theory. Studia Math. 37 (1970), 13-36. MR 0270122 | Zbl 0227.46027
[9] Wójtowicz M.: The Sobczyk property and copies of $\ell_\infty$ in locally convex-solid Riesz spaces. Arch. Math. 75 (2000), 376-379. MR 1785446

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