I have a tautological question about proving that $\text{Supp}(\tau) \subset K$, where $\tau$ is a bounded linear functional on $C(X)$, $X$: Compact Hausdorff.
We can proceed by showing that if $f \in C(X)$ such that $f|_{K}=0$, then $\tau(f)=0$. Otherwise, we can prove that if $x \not\in K$, then $x \not\in \text{Supp}(\tau)$. Towards this end, we can proceed by finding a continuous function $g$ on $C(X)$ with $x \in\text{Supp}(g)$ and $\tau(g)=0$. I was wondering if there is any fault in my reasoning for the second approach.
Thanks for the help!!
I think problem with your argument is that you're not realizing that $K \subset C(X)$. So in your first argument when you say $f\in C(X)$ what do you mean by saying $f|_K$, K is not a subset of $X$.
Your second argument seems right. So to prove your claim you need to prove that if $f\notin K$ then $f\notin Supp(\tau)$