Basic Analysis – Gently Done Topological Vector Spaces

7. Locally Convex Topological Vector Spaces
We have seen that a normed space is an example of a topological vector space.
A further class of examples is got courtesy of seminorms rather than norms. Let
P = {p α : α ∈ I} be some family of seminorms on a vector space X. As a possible
generalization of the idea of an open ball in a normed space, we consider the set
V(x 0 ,p 1 ,p 2 ,...,p n ;r) = {x ∈ X : p i (x−x 0 ) < r, 1 ≤ i ≤ n}
where x 0 ∈ X, r > 0 and p 1 ,p 2 ,...,p n is a finite collection of seminorms in P. We
will use such sets to construct a topology on X in much the same way that open
balls are used to determine a topology in a normed space. Indeed, if the family
of seminorms contains just one member, which happens to be a norm, then our
construction will give precisely the usual norm topology on X. Notice that
V(x 0 ,p 1 ,p 2 ,...,p n ;r) = x 0 +V(0,p 1 ,p 2 ,...,p n ;r).
The idea is to define local neighbourhood bases at each point of X via these sets.
Thus we are just using translates of the collection centred at the origin and so we
might hope that, indeed, we end up with a vector topology.
Theorem 7.1 Let X be a vector space over K and let P be a family of seminorms
on X. For each x ∈ X, let N x denote the collection of all subsets of X of the
form V(x,p 1 ,p 2 ,...,p n ;r), with n ∈ N, p 1 ,...,p n ∈ P and r > 0. Let T be the
collection of subsets of X consisting of ∅ together with all those subsets G of X
such that for any x ∈ G there is some U ∈ N x such that U ⊆ G. Then T is a
topology on X compatible with the vector space structure and the sets N x form
an open local neighbourhood base at x. Furthermore, each seminorm p ∈ P is
continuous. (X,T) is Hausdorff if and only if the family P is separating, that is,
for any x ∈ X with x �= 0, there is some p ∈ P such that p(x) �= 0.
Proof Evidently X ∈ T, and it is clear that the union of any family of elements
of T is also a member of T. We shall show that if A,B ∈ T then A ∩ B ∈ T.
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