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

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Consequences of the Hahn-Banach theorems Theorem (Theorem 3.5) Let X be a locally convex topological vector space and M ⊆ X a subspace. If x0 ∈ X is NOT in the closure M there is a functional l ∈ X ∗ such that l(M) = {0} and l (x0) = 1. Proof. Apply the Hahn-Banach separation theorem, Theorem 3.4 b), with A = M and B = {x0} to obtain L ∈ X ∗ and γ ∈ R such that Re L(M) < γ < Re L(x0). Since 0 ∈ M we conclude that γ > 0. Set l = L(x0) −1 L. Klaus Thomsen 7. Consequences of the Hahn-Banach theorems
Consequences of the Hahn-Banach theorems Theorem (Theorem 3.6) Let X be a locally convex topological vector space and M ⊆ X a subspace. If f : M → Φ is a continuous linear functional on M, there is an l ∈ X ∗ such that l|M = f . 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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