Let $E$ is a closed bounded set of $\mathbb{R}$. Is $C（E$ a dual of any linear norm space?
2026-03-28 20:57:54.1774731474
Is C（E）a dual of any linear norm space?
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Let us assume that the question requires $E$ to be a locally compact Hausdorff space.
It is well-known that $C(E)$ with the supremum norm is a C$^*$-algebra. Sakai proved in 1971 that a C$^*$-algebra is a dual precisely when it is a von Neumann algebra. In terms of $E$, this means that it has to be compact and extremely disconnected: the closure of every open subset is open.
The only easy examples of this are the cases where $E$ is finite and when $E=\beta\mathbb N$ (the Stone-Čech compactification of the naturals).
In particular, $C[0,1]$ is not a dual.