If $X$ is a Banach space, is it true that $X^{*}$ must contain an infinite dimensional reflexive subspace? I know of Gowers' example of a Banach space not containing $c_0$, $l_1$, or reflexive, but I cannot tell if it is a dual space.
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4$\begingroup$ $\ell_1$ is a dual space and contains no infinite dimensional reflexive subspace. $\endgroup$– Bill JohnsonCommented May 25, 2016 at 20:15
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$\begingroup$ Ah, of course, thank you. Is it known whether Gowers' space not containing $c_0$, $l_1$, or reflexive is a dual space? $\endgroup$– MarkusCommented May 25, 2016 at 20:23
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2$\begingroup$ Yes, it has a boundedly complete basis, so is isomorphic to a dual space. This is mentioned in the first paragraph of $$ $$ Argyros, Spiros A.; Arvanitakis, Alexander D.; Tolias, Andreas G. Saturated extensions, the attractors method and hereditarily James tree spaces. Methods in Banach space theory, 1–90, London Math. Soc. Lecture Note Ser., 337, Cambridge Univ. Press, Cambridge, 2006. $\endgroup$– Bill JohnsonCommented May 25, 2016 at 20:42
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$\begingroup$ That answers my questions, than you for the reference. $\endgroup$– MarkusCommented May 25, 2016 at 20:47
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