Abstract.
In a recent paper, Ghenciu and Lewis studied strong Dunford-Pettis sets and made the following two assertions:
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(1)
The Banach space X * contains a nonrelatively compact strong Dunford-Pettis set if and only if ℓ∞ ↪ X *.
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(2)
If c 0 ↪ Y and H is a complemented subspace of X so that H * is a strong Dunford-Pettis space, then W(X, Y) is not complemented in L(X, Y).
While the statements are correct, the proofs are flawed. The difficulty with the proofs is discussed, and a fundamental result of Elton is used to establish a simple lemma which leads to quick proofs of both (1) and (2).
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References
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I Ghenciu P Lewis (2005) ArticleTitleStrong Dunford-Pettis sets and spaces of operators Monatsh Math 144 275–284 Occurrence Handle1084.46013 Occurrence Handle10.1007/s00605-004-0295-7
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The online version of the original article can be found at 10.1007/s00605-004-0295-7.
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Ghenciu, I., Lewis, P. Strong Dunford-Pettis sets and spaces of operators (Monatsh. Math. 144, 275–284 (2005)). Mh Math 151, 341–343 (2007). https://doi.org/10.1007/s00605-007-0450-z
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DOI: https://doi.org/10.1007/s00605-007-0450-z