In a proof of the representation of linear functionals of topological vector space

Question

I'm reading the proof of the following theorem in a note on functional analysis:

Here $p_F$ is defined as $p_F(x)=\max_{y\in F}|\langle x,y\rangle|$. Could anyone show me why the underscored sentences are true? The theorem is stated right after the definitions of dual pairs and weak topology in the chapter of locally convex spaces. I'm not sure if I'm missing something which is implicitly used.

Answer 1

By definition, $\sigma(X,Y)$ is generated by the semi-norms $p_F$. Moreover, a linear functional $\varphi$ on a l.c.s. generated by the directed family $\mathscr P$ of semi-norms is continuous if and only if there are $p\in\mathscr P$ and $c>0$ such that $|\varphi|\le cp$.

The functional $\psi$ on $T(X)$ is defined by $\psi(T(x))=\varphi(x)$ -- you just have to check that this is well-defined, i.e. $T(x)=T(\xi)$ implies $\varphi(x)=\varphi(\xi)$ (which follows from $\varphi(x)=0$ if $\langle x,y_j\rangle=0$ for all $j$ applied to $x-\xi$).