Basic Analysis – Gently Done Topological Vector Spaces

7: Locally Convex Topological Vector Spaces 71
Next, we observe that (i) implies that there is some neighbourhood U of 0 such
that ρ(U) ⊆ {t ∈ K : |t| < 1}, i.e., |ρ(x)| < 1 whenever x ∈ U and so (iii) holds.
Conversely, if (iii) holds, there is some neighbourhood V of 0 and some constant
C > 0 such that |ρ(x)| < C for any x ∈ V. It follows that for any ε > 0, |ρ(x)| < ε
V and so ρ is continuous at 0, which is (i).
whenever x ∈ C
ε
Finally, we show that (i) implies (iv). So suppose that ρ is continuous at 0.
Then there is a neighbourhood U of 0 such that ρ(U) ⊆ {t ∈ R : |t| < 1}, that
is, ρ(x) < 1 whenever x ∈ U. But there are p1 ,...,p m in P and r > 0 such that
V(0,p 1 ,...,p m ;r) ⊆ U, and therefore ρ(x) < 1 for all x ∈ V(0,p 1 ,...,p m ;r).
Let s(x) = p1 (x) + ··· + pm (x) and suppose x is such that s(x) �= 0. Then
rx/s(x) ∈ V(0,p 1 ,...,p m ;r) and so ρ(rx/s(x)) < 1, that is,
ρ(x) < 1 �
p1 (x)+···+p m (x)
r
� .
On the other hand, if s(x) = 0, then s(nx) = 0 for any n ∈ N. Hence nx ∈
V(0,p 1 ,...,p m ;r) and so ρ(nx) = nρ(x) < 1 for all n ∈ N. This forces ρ(x) = 0
and we conclude that, in any event,
for all x ∈ X.
ρ(x) ≤ 1 �
p1 (x)+···+p m (x)
r
�
Corollary 7.9 Suppose that Λ : X → K is a linear functional on a topological
vector space (X,T), where T is the vector space topology determined by a family
P of seminorms on X. The following statements are equivalent.
(i) Λ is continuous at 0.
(ii) Λ is continuous.
(iii) There is a finite set of seminorms p 1 ,...,p m in P and a constant C > 0
such that
|Λ(x)| ≤ C(p 1 (x)+···+p m (x)) for x ∈ X.
If F is a family of linear functionals on X such that P is the collection P = {|ℓ| :
ℓ ∈ F}, then (i), (ii) and (iii) are equivalent to the following.
(iv) There is a finite set ℓ 1 ,...,ℓ m of members of F and s 1 ,...,s m ∈ K such
that
Λ(x) = s 1 ℓ 1 (x)+···+s m ℓ m (x) for x ∈ X.
Proof We know already that (i) and (ii) are equivalent, by Proposition 6.19. For
x ∈ X, set ρ(x) = |Λ(x)|. Then ρ is a seminorm on X. To say that ρ is continuous