In the book Introduction to Functional Analysis written by A. E. Taylor there are the following theorems:
Theorem 1. Suppose that $X$ is a linear space and that $\mathscr{U}$ is a nonempty family of nonempty subsets of $X$ with the properties:
$(i)$ Each member of $\mathscr{U}$ is balanced, convex and absorbing;
$(ii)$ If $U\in \mathscr{U}$ there exists some $\alpha$ such that $0<\alpha\leq 1/2$ and $\alpha U\in\mathscr{U}$;
$(iii)$ If $U_1$ and $U_2$ are in $\mathscr{U}$ there exists $U_3\in\mathscr{U}$ such that $U_3\subset U_1\cap U_2$
$(iv)$ If $U\in \mathscr{U}$ there exists $V\in\mathscr{U}$ such that $x+V\subset U$.
Then there is a unique topology for $X$ such that $X$ is alocally convex topological vector space with $\mathscr{U}$ as a base at $0$.
Using this we might prove:
Theorem 2. Let $P$ be a nonempty family of seminorms defined on the linear space $X$. For each $p\in P$ let $V(p)$ be the set $\{x: p(x)<1\}$. Let $\mathscr{U}$ be the family of all finite intersections $$r_1V(p_1)\cap r_2V(p_2)\cap \ldots \cap r_n V(p_n),\ r_k>0,\ p_k\in P$$ Then $\mathscr{U}$ satisfies the conditions $(i)-(iv)$ of the previous theorem.
I have a doubt concerning the second theorem:
Question: It is not clear to me the definition of the set $$r_1V(p_1)\cap r_2V(p_2)\cap \ldots \cap r_n V(p_n),\ r_k>0,\ p_k\in P.$$ Does he mean $$U\in\mathscr{U}\Leftrightarrow \exists\ p_1, \ldots, p_k\in P\ \textrm{and}\ r_1, \ldots, r_k>0; U=\bigcap_{i=1}^k r_iV(p_i)?$$