$C(K)^*$ is not separable

Question

I found the following problem

Problema.- Fixed a compact metric space $(K,d)$ consider the Banach space $C(K)=\{f:K\to\mathbb R: f\text{ is continuous}\}$. Then

1. Calculate $\|G_x\|_*$, where $G_x(f)=f(x)$.

$\|G_x\|_*=\displaystyle\sup_{\|f\|=1}|G_x(f)|=\sup_{\|f\|=1}|f(x)|\leq 1$. Then if we take $f_0\equiv 1$, then $\|G_x\|_*=\displaystyle\sup_{\|f\|=1}|G_x(f)|>|f_0(x)|=1$. Hence $\|G_x\|_*=1.$

2.Calculate the distance $\|G_x-G_y\|_*$, for each pair of points $x,y\in K$.

$\|G_x-G_y\|_*=\displaystyle\sup_{\|f\|=1}|(G_x-G_y)(f)|=\sup_{\|f\|=1}|f(x)-f(y)|\leq 2$. But I have not achieved the other inequality, it would be simpler if K were convex since it could take a continuous function such that $f(x)=1$ and $f(y)=-1$, but this is not the case.

3. Prove that if $K$ it is not countable, then

• $C(K)^*$ is not separable.
• $C(K)^{**}$ is not separable.
• $C(K)$ is not reflexive.

In that items I don't know how to attack it

Answer 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.

Answer 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.