ABSTRACT
We call the weakest topology that makes each member of a family P of seminorms on a vector space continuous a “seminorm topology,” a topic we first discussed in Example 4.5.4. The local convexity of a seminorm topology stems from the fact that the open ball Vp = {x ∈ X : p(x) < 1} determined by a seminorm p is convex, a consequence of the fact that p satisfies the triangle inequality. Indeed, a topology is locally convex iff it is generated by a family of seminorms. Two prominent seminorms are:
Let C(R,R) be the linear space of all continuous maps of R into R. For each n ∈ N, let pn(x) = sup |x([−n, n])| , x ∈ C (R,R). Each pn is a seminorm and the topology determined by P = {pn : n ∈ N} on C(R,R) is called the compact-open topology.
