7. Consequences of the Hahn-Banach theorems - Aarhus Universitet
7. Consequences of the Hahn-Banach theorems - Aarhus Universitet
7. Consequences of the Hahn-Banach theorems - Aarhus Universitet
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<strong>Consequences</strong> <strong>of</strong> <strong>the</strong> <strong>Hahn</strong>-<strong>Banach</strong> <strong>the</strong>orems<br />
Pro<strong>of</strong>.<br />
Because Λ00 is linear, it follows easily that Λ00(B) is convex and<br />
balanced since B is. It is <strong>the</strong>n easy to see (!!) that <strong>the</strong>re is an<br />
r ≥ 0 such that<br />
{z ∈ C : |z| < r} ⊆ Λ00(B) ⊆ {z ∈ C : |z| ≤ r}.<br />
Note that r ≤ 1 by (1), and that<br />
|Λ00(b)| ≤ r ≤ 1 < ReΛ00(x0) (2)<br />
for all b ∈ B.<br />
Choose α ∈ C such that |α| = 1 and αΛ00(x0) = |Λ00(x0)|. Set<br />
Λ = αΛ00, note that<br />
for all b ∈ B.<br />
Λ(x0) ≥ ReΛ00(x0) > 1 ≥ |Λ00(b)| = |Λ(b)|<br />
Klaus Thomsen <strong>7.</strong> <strong>Consequences</strong> <strong>of</strong> <strong>the</strong> <strong>Hahn</strong>-<strong>Banach</strong> <strong>the</strong>orems