I am reading a paper on Riesz Representation Theorem, (https://www.ams.org/journals/proc/1970-024-03/S0002-9939-1970-0415386-2/S0002-9939-1970-0415386-2.pdf this one in particular), and there is a part in introduction that I don't understand, that I copied underneath:
Let $S$ denote the Borel sets, i.e., the $\sigma$-ring generated by closed subsets of $H$ ($H$ being a compact Hausdorff space). If $e\in S$ we denote the characteristic function of $e$ by $\chi_e$. If $e\in S$ and $x\in E$ ($E$ being a locally convex topological vector space) we think of $\chi_e$ as an element of $C''(H, E)$ by the following identification: if $f' \in C(H, E)$ then $\phi$ defined by $\langle f \cdot x, f' \rangle$ for $f \in C(H)$ is in $C'(H)$. Thus there exists a unique regular Borel measure $\mu_{(x, f')}(e)$ such that $\phi (f)=\int fd\mu_{(x, f')}$. Now define $\langle \chi_e \cdot x, f' \rangle =\mu_{(x, f')}(e)$. $\chi_e \cdot x \in C''(H, E)$ since it is in the bipolar of the bounded set $A= \{f\cdot x; \|f\|\le 1\}$. For if $f' \in A ^{\circ}$ then $|\langle \chi_e \cdot x, f' \rangle|=|\mu_{(x, f')}| \le \| \mu_{(x, f')}(e)\|$ (variation of $\mu_{(x, f')}$). But: $$ \| \mu_{(x, f')}(e)\| = \|\phi\|= sup_{\|f\|\le 1}|\langle f, \phi \rangle | = sup_{\|f\|\le 1} |\langle f \cdot x, f' \rangle | \le 1 $$ since $f' \in A^{\circ}$. Thus $\chi_e \cdot x \in C''(H, E)$.
So this is the paragraph, and what I didn't understand is the the part where $\chi_e \cdot x \in C''(H, E)$ since it is in the bipolar of the bounded set A - why is it in the bipolar, and why is A a bounded set at all?