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

Consequences of the Hahn-Banach theorems
Proof.
Because Λ00 is linear, it follows easily that Λ00(B) is convex and
balanced since B is. It is then easy to see (!!) that there is an
r ≥ 0 such that
{z ∈ C : |z| < r} ⊆ Λ00(B) ⊆ {z ∈ C : |z| ≤ r}.
Note that r ≤ 1 by (1), and that
for all b ∈ B.
|Λ00(b)| ≤ r ≤ 1 < ReΛ00(x0) (2)
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems