Let $\Omega$ be a compact metric space and $C(\Omega,\mathbb{R})$ the space of borel continuous real valued functions. I would like to know if there is any real Banach space $V$ such that its dual space (topological)is exactly $C(\Omega,\mathbb{R})$.
My main interest is when $\Omega$ is an infinite cartesian product as for example, $\Omega=E^{\mathbb{Z}^d}$, where $E$ is a compact metric space. If the answer for this case is also negative are we in a better situation if $E$ is a finite set ?
Thanks for any comment or reference.
Edition. Remove the superfluous hypothesis pointed out by Philip.