This is what I have constructed so far.
Let $V$ be a vector space and $\{m_i\}_{i \in I}$ be a separating family of semi-norms on $V$. Define $$f_{i, y}(x) := m_i(x - y)$$We can show that the collection of $f_{i, y}$ is a separating family on $V$. Suppose that $x \neq y$. Hence, $x - z \neq y - z$. There exists $i \in I$ such that $$f_{i, z}(x) = m_i(x - z) \neq m_i(y - z) = f_{i, z}(y)$$We can consider the initial topology i.e. the weakest topology on $X$ that makes every function in our collection $\{f_{i, y}\}_{i \in I, y \in V}$ continuous.
He then states that sets of this form a subbasis for the neighborhood filter $\mathcal{O}(x)$.
$$\{y \in V: |m_i(y - z) - m_i(x - z)| < \epsilon\} $$ I understand that these sets will be in the subbasis of my weakest topology. However, shouldn't there be more sets in my subbasis that contain $x$? We can view $$ \{y \in V: |m_i(y - z) - m_i(x - z)| < \epsilon\} = f_{i, z}^{-1}(B(m_i(x - z), \epsilon))$$ Couldn't we have some open ball $B$ in $\mathbb{R}_{\geq 0}$ that's center isn't $m_i(x - z)$? The thing that I was having trouble on understand was that Pedersen doesn't seem to be using on open sets $U$ in $\mathbb{R}_\geq$ where $x \in f_{i, z}^{-1}(U)$.
I felt that I understood the rest of Pedersen's explanation of the construction of a locally convex topological vector space from a collection of semi-norms.
If $x\in f^{-1}_{i, z}(U)$, then $f_{i, z}(x)\in U$. Since $U$ is open, there exists $\varepsilon>0$ such that $I=(f_{i, z}(x)-\varepsilon, f_{i, z}(x)+\varepsilon)\subset U$, and so $f^{-1}_{i, z}(I)\subset f^{-1}_{i, z}(U)$. But $f^{-1}_{i, z}(I)=\{y\in V:\, |f_{i, z}(y)-f_{i, z}(x)|<\varepsilon\}$. So it is enough to consider sets of the type $f^{-1}_{i, z}(I)$. Does this answer your question?