POTW Weak Topology on an Infinite-Dimensional Hilbert Space

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Prove that the weak topology on an infinite-dimensional Hilbert space is non-metrizable.

 

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Spoiler

By way of contradiction, suppose that there exists a metrizable, infinite-dimensional Hilbert space $H$. Let $d$ be a metric on $H$ inducing the weak topology. For every $n \ge 1$, the origin belongs to the weak closure of the sphere in $H$ of radius $n$. So there is a sequence $(x_n)\subset H$ with $\|x_n\| = n$ such that $d(x_n,0) < n^{-1}$. Thus $x_n$ weakly converges to $0$, even though $(x_n)$ is unbounded. ($\rightarrow\leftarrow$)