Let $X$ be a Banach space and suppose that the weak topology on $X^*$ agrees with the weak* topology on $X^*$. Must $X$ be reflexive?
To prove the contrapositive, it will suffice to assume that $X$ is not reflexive and construct a sequence $\phi_n \in X^*$ such that $\phi_n(x) \rightarrow \phi(x)$ for each $x\in X$, but $\lambda(\phi_n)\not\rightarrow \lambda(\phi)$ for some bounded linear fucntional $\lambda$ on $X^*$. However, I have been unable to do so. Does anyone have any ideas?