If X = E* is the dual space of a Banach space E, then one can show that the weak-* topology on X is not metrizable, similarly to the proof of Proposition 2 above (the unit ball of X, however, will be metrizable when E is separable).

So if you want such an example, you need E to be an infinite dimensional normed space which is not complete. If you take the simplest example (say, finitely supported sequences with a reasonable norm) then it is not hard to show that the weak-* topology on X=E* determined by E will be metrizable. ]]>

weak* topology of X· is metrizable. ]]>

The proof given in Arveson’s book that I mention at the top of the post (and similar proofs appear in many books) uses complex function theory, but avoids the holomorphic functional calculus. One might argue that the holomorphic functional calculus lurks in the background of such proofs.

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