Let $S$ be compact and Hausdorff and $C(S)$ be its space of continuous complex functions. When $C(S)$ is endowed with the $\sup$ norm, its dual is well known.
Since this topology is too strong for my needs, are the duals of $C(S)$ known when this is endowed with the topology of pointwise convergence and, respectively, with the weak topology?
In fact, I need this result not for itself, but rather for finding the preduals of $C(S)$ in these two topologies (using next that, in general, $X \simeq \big( X^*, \sigma^* (X^*, X) \big) ^*$).
The weak topology has the property that the dual is the same as the dual with respect to the original topology, if $(E,\tau)$ is a topological vector space, and $E^\ast$ its (topological) dual, then
$$(E,\sigma(E,E^\ast))^\ast = E^\ast.$$
For the topology of pointwise convergence on $C(S)$, the dual is easily described: if $\lambda \colon C(S) \to \mathbb{C}$ is continuous with respect to the topology of pointwise convergence, then there are $s_1,\dotsc, s_n \in S$ and $c_1,\dotsc, c_n \in \mathbb{C}$ with
$$\lambda(f) = \sum_{k = 1}^n c_k\cdot f(s_k),$$
the dual of $C(S)$ in the topology of pointwise convergence is the span of the evaluation functionals. That follows since $C(S)$ is then a dense linear subspace of $\mathbb{C}^S$ in the product topology.