Let $X$ be the vector space of all complex functions on $[0,1]$, topologized by the family of seminorms $p_{x}(f)=|f(x)|$, $0\le x\le1$. This topology is called the topology of pointwise convergence. There is Theorem 1.37 in Rudin's Functional analysis which gives us the base for this topology. Am I right that these sets are $V_n=\{f:[0,1]\to\mathbb{C}\,|\,|f(x)|<\frac{1}{n},x\in J_n\}$, where $J_n$ is a finite subset of $[0,1]$? How can we establish equivalency between convergence in this topology and the standard pointwise convergence? Any help would be highly appreciated. I am confused about these sets $J_n$. They are finite (because of finite intersections from the theorem), but this is a countable base, so their union cannot be the whole $[0,1]$. I am not understanding something well here...
Theorem 1.37: Suppose $P$ is a separating family of seminorms on a vector space $X$. Associate to each $p\in P$ and to each positive integer $n$ the set $V(p,n)=\{x:p(x)<\frac{1}{n}\}$. Let $B$ be the collection of all finite intersections of the sets $V(p,n)$. Then $B$ is a convex balanced base for a topology $\tau$ on $X$, which turns $X$ into a localy convex space such that every p is continuous...