Let $X$ be a subspace of a space with unconditional basis. Show that if $X$ contains no copy of $c_0$ or $l_1$ then $X$ is reflexive.
We know that a Banach space is reflexive iff it has a basis which is both boundedly complete and shrinking.
If we have $X$ as a space with unconditional basis then since $X$ doesn't have a copy of $c_0$ so the basis is boundedly complete and since $X$ doesn't have a copy of $l_1$, the basis is shrinking. Now since basis is both shrinking and boundedly complete $X$ is reflexive.
But the problem is that I don't know if we can say $X$ has unconditional basis
Any help is appreciated! Thank you.
This is Exercise 3.6 in Kalton and Albiac's Topics in Banach Space theory.
The result, and its proof can be found in the paper On bases and unconditional convergence of series in Banach spaces, C. Bessaga, A. Pełczyński, Studia Mathematica 17 (1958), 151-164. This paper will henceforth be denoted by $[1]$, and can be found here.
The result follows immediately from the main results of this paper:
1) Since $X$ contains no copy of $c_0$, it must be weakly sequentially complete (every weakly Cauchy sequence in $X$ is weakly convergent).
and
2) Since $X$ does not contain a copy of $\ell_1$, every bounded sequence in $X$ has a weakly Cauchy subsequence.
Combining these, it follows that every bounded sequence in $X$ has a weakly convergent subsequence. By the Eberlein–Šmulian theorem, then, the closed unit ball of $X$ is weakly compact; and so, $X$ is reflexive. $${}$$ The proof of 1) given in $[1]$ is somewhat intimidating; but I imagine a simplified proof of 1) (which is likely what Kalton and Albiac had in mind) is contained in the reference of the James' paper mentioned in the comments.
1) follows from the fact that $X$ must have property $(u)$, as defined in Definition 3.5.1 of Kalton/Albiac. It follows from the definition of property $(u)$ that if $X$ had a weakly Cauchy sequence that was not weakly convergent, then $X$ would have a WUC (weakly unconditionally Cauchy) series that is not convergent. This would imply that $X$ must contain a copy of $c_0$ (Theorem 2.4.11 in Kalton/Albiac).
2) follows immediately from Rosenthal's $\ell_1$-theorem: a Banach space $X$ does not contain a copy of $\ell_1$ iff every bounded sequence in $X$ has a weakly Cauchy subsequence. This is a bit of a sledgehammer, of course; in [1], the $\ell_1$-theorem is proved for a Banach space $X$ that is a subspace of a space with an unconditional basis (this proof can be reproduced using results from Kalton/Albiac prior to this exercise).