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Embedding \(c_0\) in the space of Pettis integrable functions. (English) Zbl 0963.46025
Quaest. Math. 21, No. 3-4, 261-267 (1998); correction ibid. 29, No. 1,. 133-134 (2006).
Let \(P_1(\mu,X)\) (resp. \(P(\mu,X))\) be the normed space, generally noncomplete, of \(\mu\)-measurable (resp. weakly \(\mu\)-measurable) Pettis integrable functions defined on a probability space \((S,\Sigma, \mu)\) with values in the Banach space \(X\). The author shows that if \(\mu\) has infinite range and \(X\) contains \(c_0\), then so does \(P_1(\mu, X)\) which furthermore is complemented in \(P(\mu, X)\). Conversely, if \(P_1(\mu, X)\) contains a copy of \(c_0\), so does \(X\). This generalizes a result of S. Diaz, A. Fernandez, M. Florencio and P. F. Paul [Quest. Math. 16, No. 1, 61-66 (1993; Zbl 0830.46014)].

46G10 Vector-valued measures and integration
28B05 Vector-valued set functions, measures and integrals
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46B20 Geometry and structure of normed linear spaces
Full Text: DOI
[1] Cembranos P., Proc. Amer. Math. Soc. 91 pp 556– (1984)
[2] Drewnowski L., Proc. Amer. Math. Soc. 109 pp 747– (1990)
[3] DOI: 10.1080/16073606.1993.9631715 · Zbl 0830.46014 · doi:10.1080/16073606.1993.9631715
[4] Diestel J., Vector Measures (1977)
[5] Emmanuele G., Comment. Math. Prace Mat.
[6] DOI: 10.1007/BF01455966 · Zbl 0525.46022 · doi:10.1007/BF01455966
[7] Hoffmann-Jrgensen J., Studia Math. 52 pp 159– (1974)
[8] Kwapień S., Studia Math. 52 pp 187– (1974)
[9] Lindenstrauss J., Classical Banach Spaces I (1977) · Zbl 0362.46013
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