Basic Analysis – Gently Done Topological Vector Spaces

114 Basic Analysis
and this is separating. The translation invariant metric d induced by the norm, as
above, is given by d(x,y) = �x−y�/(1+�x−y�). Clearly d and the usual metric
(also translation invariant) given by the norm, namely, (x,y) ↦→ �x−y� have the
same Cauchy sequences.
Theorem 10.6 Let(X,T)beafirstcountable, Hausdorfflocallyconvextopological
vector space. Then there is a countable family P of seminorms on X such that
T = T P , the topologydetermined by P. In particular, T is induced by a translation
invariant metric.
Proof By hypothesis, there is a countable neighbourhood base at 0. However, any
neighbourhood of 0 in a locally convex topological vector space contains a convex
balanced neighbourhood so there is a countable neighbourhood base {U n } n∈N at
0 consisting of convex balanced sets. For each n, let p n denote the Minkowski
functional associated with U n . The argument of Theorem 7.25 shows that T = T P ,
where P is the family {p n : n ∈ N}.
For any x ∈ X with x �= 0, there is some n such that x /∈ U n , since T is
Hausdorff, by hypothesis. Hence p n (x) ≥ 1 and so P is separating.
The topology T = T P is given by the translation invariant metric
for x,y ∈ X.
d(x,y) =
∞�
n=1
1
2 n
p n (x−y)
1+p n (x−y) ,
Corollary 10.7 If (X,T) is a metrizable locally convex topological vector space,
then T is determined by a countable family of seminorms on X.
Proof If (X,T) is metrizable, T is Hausdorff and first countable. The result now
follows from the theorem.
Theorem 10.8 Let (X,T) be a first countable separated locallyconvex topological
vector space such that every Cauchy sequence (with respect to T) converges. Then
(X,T) is a Fréchet space.
Proof From the above, we see that (X,T) is a topological vector space whose
topology is determined by a countable family P of seminorms. The convergence
of Cauchy sequences with respect to T is equivalent to their convergence with
respect to the translation invariant metric d associated with the countable family
of seminorms, P. In other words, (X,T) is equal to (X,d), as topological vector
spaces, and (X,d) is complete, i.e., (X,T) is a Fréchet space.
Department of Mathematics King’s College, London