$l_1$ has no infinite dimensional subspace that is reflexive.

Question

How to show that $l_1$ does not contain an infinite dimensional subspace that is reflexive.

Answer 1

See Schur's theorem and Eberlein-Smulian Theorem. Suppose that $Y \le \ell_1$ is a subspace, and let $B_Y$ be the closed unit ball of $Y$, i.e. $B_X \cap Y$. Then \begin{align*} Y \text{ is reflexive} &\iff B_Y \text{ is weakly compact} \\ &\iff B_Y \text{ is sequentially weakly compact} &\text{(Eberlein-Smulian)} \\ &\iff B_Y \text{ is (sequentially) norm compact} &\text{(Schur)} \\ &\iff Y \text{ is finite-dimensional}. \end{align*}