I have a question about the proof of Theorem 8.2.10 from "Topological Vector Spaces" by Narici & Beckenstein.
Let $(X,Y,b)$ be a pairing of topological vector spaces over the field $\mathbb{K}$. If $g$ is a $\sigma(X,Y,b)$-continuous functional on $X$, then there exists some $y\in Y$ such that $g=b(\cdot,y)$.
They start with mentioning that continuity of a functional $f$ on a topological vector space $X$ is equivalent to $|f|\leq p$ for some continous seminorm $p$ on $X$. Then they continue: This means, that for a $\sigma(X,Y,b)$-continuous linear functional $g$, there exist $y_1,\dots, y_n\in Y$ such that $$|g|\leq\max_i |b(\cdot,y_i)|.$$ I just don't see how one implies the other.