In the proof of the following theorem:
if $X$ is a separable Fréchet space, then $\mathcal{E}(X)=\mathcal{B}(X)$.
Where $\mathcal{E}(X)$ is the $\sigma$-algebra generated by the cylindrical sets on $X$ and $\mathcal{B}(X)$ is the Borel $\sigma$-algebra.
They start by taking $(x_n)$ a dense sequence in $X$, and a family of seminorms $(p_k)$ which defines the topology of $X$. The part of the proof that I didn't get is the following:
By the Hahn-Banach theorem for every $n$ and $k$ there is $l_{n,k}\in X^*$ such that $p_k(x_n)=l_{n,k}(x_n)$ and $\sup\{l_{n,k}(x):\ p_k(x)\leq 1\}=1$. As a consequence, for every $x\in X$ and $k\in\mathbb{N}$ we have $p_k(x)=\sup_n\{l_{n,k}(x)\}$.
Questions:
*on which linear application did they apply the Hahn-Banach theorem ?
*what did they use to obtain $p_k(x)=\sup_n\{l_{n,k}(x)\}$ as a consequence ?
Thank you for your time.