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

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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 |Λ00(b)| ≤ r ≤ 1 < ReΛ00(x0) (2) for all b ∈ B. Choose α ∈ C such that |α| = 1 and αΛ00(x0) = |Λ00(x0)|. Set Λ = αΛ00, note that for all b ∈ B. Λ(x0) ≥ ReΛ00(x0) > 1 ≥ |Λ00(b)| = |Λ(b)| Klaus Thomsen 7. Consequences of the Hahn-Banach theorems
Weak vector space topologies Let X be a real or complex vector space, and M a collection of lineat functionals X → Φ. Klaus Thomsen 7. Consequences of the Hahn-Banach theorems
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7. Consequences of the Hahn-Banach theorems - Aarhus Universitet

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