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

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Weak vector space topologies Proof. Conversely, assume that M separates the points of X. Let x0,y0 ∈ X, x0 = y0. There is then a functional l ∈ M such that l(x0) = l(y0). Set ǫ = 1/2 |l(x0) − l(y0)| > 0, and note that Since x0 ∈ {y ∈ X : |l(y) − l(x0)| < ǫ} , y0 ∈ {y ∈ X : |l(y) − l(y0)| < ǫ} . {y ∈ X : |l(y) − l(y0)| < ǫ} ∩ {y ∈ X : |l(y) − l(x0)| < ǫ} = ∅, the proof is complete. Klaus Thomsen 7. Consequences of the Hahn-Banach theorems
Weak vector space topologies Theorem Theorem 3.10. Let X be a vector space and X ′ a separating vector space of linear functional on X. Then the X ′ -topology makes X a locally convex topological vector space whose dual is X ′ . 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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