𝐶(𝐾,𝐸) contains a complemented copy of 𝑐₀

Cembranos, Pilar

doi:10.1090/S0002-9939-1984-0746089-2

$C(K, E)$ contains a complemented copy of $c_{0}$
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Proc. Amer. Math. Soc. 91 (1984), 556-558 Request permission

Abstract:

Let $E$ be a Banach space and let $K$ be a compact Hausdorff space. We denote by $C(K,E)$ the Banach space of all $E$-valued continuous functions defined on $K$, endowed with the supremum norm. We prove in this paper that if $K$ is infinite and $E$ is infinite-dimensional, then $C(K,E)$ contains a complemented subspace isomorphic to ${c_0}$.

References

C. Bessaga and A. Pełczyński, On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151–164. MR 115069, DOI 10.4064/sm-17-2-151-164
Philip C. Curtis Jr., A note concerning certain product spaces, Arch. Math. 11 (1960), 50–52. MR 111008, DOI 10.1007/BF01236906
J. Diestel and J. J. Uhl Jr., Vector measures, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis. MR 0453964, DOI 10.1090/surv/015
Surjit Singh Khurana, Grothendieck spaces, Illinois J. Math. 22 (1978), no. 1, 79–80. MR 458144
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056, DOI 10.1007/978-3-642-66557-8
A. Nissenzweig, $W^{\ast }$ sequential convergence, Israel J. Math. 22 (1975), no. 3-4, 266–272. MR 394134, DOI 10.1007/BF02761594
A. Pełczyński, Banach spaces on which every unconditionally converging operator is weakly compact, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 10 (1962), 641–648. MR 149295
Elias Saab and Paulette Saab, A stability property of a class of Banach spaces not containing a complemented copy of $l_{1}$, Proc. Amer. Math. Soc. 84 (1982), no. 1, 44–46. MR 633274, DOI 10.1090/S0002-9939-1982-0633274-1
Andrew Sobczyk, Projection of the space $(m)$ on its subspace $(c_0)$, Bull. Amer. Math. Soc. 47 (1941), 938–947. MR 5777, DOI 10.1090/S0002-9904-1941-07593-2