$C(K)^*$ is not separable

Solution 1:

Because $K$ is compact metric, given two distinct points you can always create a continuous function that takes two values you want at those points. You can use Urysohn's Lemma or Tietze's Extension Theorem for this. These theorems let you also prescribe that $\|f\|=1$. So the distance is always 2.

Now, if $K$ is uncountable, that $C(K)^*$ is not separable follows directly from 2.

That $C(K)^{**}$ is the fact that the dual of a non-separable space has to be non-separable.

Whenever $K$ is infinite, $C(K)$ cannot be reflexive. This is consequence of the fact that on a reflexive space the unit ball is weakly compact.

Solution 2:

By Riesz Representation Theoem $C(K)^{*}$ is the space of Borel measures on $K$ with the total variation norm. If $a, b \in K, a \neq b$ then the distance from $\delta_a$ to $\delta_b$ is $2$. Since there are uncountably many elements of $C(K)^{*}$ at distance $2$ it follows that this space is not separable. If a normed linear space is not separable then its dual is not separable. From the fact that $C(K)$ is separable and $C(K)^{**}$ is not, it follows that $C(K)$ is not reflexive.