I am reading a proof regarding amenability of a group $G$ that includes the following:
Let $G$ be a group and let $X$ be a Banach space, $X^{*}$ its dual space, and $\pi$ an isometric representation of $G$ on $X$. For all $x^{*} \in X^{*} $ we define a function:
$\mathcal{F}_{x^*}:X \to \mathbb{C} \hspace{0.5cm} \mathcal{F}_{x^*}(x)=m(\varphi_{x,x^*})$ where $m$ is a left invariant mean on $G$ and $\varphi_{x,x^*} \in L^{\infty}(G)$ such that $\varphi_{x,x^*}(g)=x^*\circ(\pi(x)(g))$
Around half way through the proof it says this:
Assume for a contradiction that for some $x^*$, $\mathcal{F}_{x^*}\notin$ conv$\overline{\{\pi^{*}(g)x^* : g \in G\}}^{\omega^*}$. By the Hahn-Banach Theorem there are $a,b \in \mathbb{R}$ and a $\omega^*$-continuous linear functional of $X^{*}$, say $x_{0}\in X$ (since $X^{**}\cong X$) such that Re$((\mathcal{F}_{x^*})(x_{0}))\leq a<b\leq$ Re$(\varphi_{x_{0}, x^{*}})$
I don't understand how the Hahn-Banach Theorem implies the above. I read the link in this post to find a detailed explanation of the Hahn_Banach Theorem: Hahn–Banach theorem? but I don't see how it is being applied in this case. I would be really grateful if someone could explain it's use in this case!
This is making use of the geometric form of the Hahn-Banach theorem, sometimes called the 'separation theorem' as Pseudo coder commented. (These notes provide a fairly detailed explanation.)
Since $X^*$ is a Banach space, $\cal{F}_{x^*}$ is a point (and therefore closed) and $\mathrm{conv}\overline{\{\pi^*(g)x^* \mid g \in G\} } $ is a closed convex set we can apply the geometric form of the Hahn-Banach theorem which says that we can find a closed hyperplane $H=\{f^{**}=\alpha \}$ (for some number $\alpha$) that lies strictly between the two closed sets. In particular, there must then be a linear functional (the one defining $H$ for certain) which takes on distinct values on the two closed sets, one less than $\alpha$ and one greater than $\alpha$.
So, choose $a<\alpha$ and $b>\alpha$ and you obtain the conclusion stated in your question.