QUESTION: Let $E$ be an infinite-dimensional Banach space. Our purpose is to show that $E'$ equipped with the weak $^\star$ topology is not metrizable. Suppose, by contradiction, that there is a metric $d(x,y)$ on $E'$ that induces on $E'$ the same topology as $\sigma(E', E)$. Prove that $E'$ equipped with the weak$^\star$ topology $\sigma(E',E)$ is not metrizable.
HINT IN BREZIS' BOOK: Apply the following lemma: Assume that $x_1, x_1, \cdots, x_k, y\in E$ satisfy $$[f\in E';\langle f, x_i \rangle =0\; \forall i]\implies [\langle f, y \rangle=0].$$ Then there exist constants $\lambda_1, \lambda_2, \cdots, \lambda_k$ such that $y=\displaystyle\sum^{k}_{i=1}\lambda_ix_i$.
DOUBT: Would you give me a hint how to start this problem and where to use this lemma? I tried to do the same for weak topology, but I'm not very confident about what I wrote. So, I would like to be sure at least about this part about weak$^{\star}$ topology, then I can check and do similar proof for the first parte.
I appreciate any help. Thanks in advance.
