7. Consequences of the Hahn-Banach theorems - Aarhus Universitet

Consequences of the Hahn-Banach theorems
Theorem
(Theorem 3.7) Let X be a locally convex topological vector space.
Let B ⊆ X be a convex, balanced and closed subset. Let
x0 ∈ X \B. There is an l ∈ X ∗ such that |l(x)| ≤ 1 for all x ∈ B
while l(x0) > 1.
Proof.
It follows from Theorem 3.4 b) that there are a Λ0 ∈ X ∗ and a real
number γ ∈ R such that
ReΛ0(x0) < γ < ReΛ0(b)
for all b ∈ B.
Since 0 ∈ B we see that γ < 0. Set Λ00 = 1
γ Λ0, and note that
Re Λ00(x0) > 1 > Re Λ00(b), b ∈ B. (1)
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems