A Banach space $X$ is weakly sequentially complete (WSC) if every weakly Cauchy sequence in $X$ is weakly convergent.
I will use the following classical result:
Rosenthal's $\ell_1$ theorem: Every bounded sequence in a Banach space $X$ either contains a weakly Cauchy subsequence, or a subsequence equivalent to the $\ell_1$ basis.
It follows from the fact that a closed subspace of a WSC space is WSC, that every subpace of a WSC space is either reflexive or contains $\ell_1$. This observation, if localized, actually gives us a characterization of WSC spaces, namely:
Proposition. For a Banach space $X$ TFAE:
- $X$ is WSC;
- Every bounded sequence in $X$ has either a weakly convergent subsequence, or a subsequence equivalent to the $\ell_1$ basis;
- Every weakly precompact (i.e. not containing a sequence equivalent to the $\ell_1$ basis) bounded set in $X$ is relatively weakly compact.
It would thus feel plausible that the "nonlocalized" version of this Proposition could be true as well. Namely, I would like to know if:
Question: Let $X$ be a Banach space such that every closed subspace $Y$ of $X$ is either reflexive or contains $\ell_1$, is $X$ then weakly sequentially complete?
As mentioned above, the converse implication is true. I have, however, not managet to prove/disprove the Question above. Any help will be appreciated.
Best Answer
The article by Azimi & Hagler provides a Banach space $X$ with a basis $(e_n)$ such that