Weak topology on an infinite-dimensional normed vector space is not metrizable

Question

I've been pondering over this problem for a while now, but I can't come up with a proof or even a useful approach...

Let $X$ be am infinite-dimensional normed vector space over $\mathbb{K}$ (that is either $\mathbb{R}$ or $\mathbb{C}$). Then the weak topology $\sigma(X,X^*)$ is not metrizable, i.e. there is no metric $d$ such that the induced topology of $d$ coincides with $\sigma(X,X^*)$.

Can anyone help me with this?

Answer 1

Let $X$ be a normed space. You can show that if the weak topology of $X$ admits a countable base of open sets at $0$, then $X$ is finite dimensional:

• Prove the existence of a countable set $\{\zeta_n\}$ in $X^*$ such that every $\zeta \in X^*$ is a finite linear combination of the $\zeta_n$.
• Derive from this that $X^*$ is finite dimensional.
• Deduce that $X$ is finite dimensional.