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