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

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Consequences of the Hahn-Banach theorems Proof. By Theorem 3.5 there is a l ∈ X ∗ such that l(M0) = {0} and l(x0) = 1. Let x ∈ M. Then x − f (x)x0 ∈ M0 (since f (−f (x)x0) = f (x) − f (x)f (x0) = 0), and hence l(x − f (x)x0) = 0. It follows that l(x) = f (x). Klaus Thomsen 7. Consequences of the Hahn-Banach theorems
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. 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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