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

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Weak vector space topologies Lemma The M-topology is Hausdorff if and only if M separates the points of X. Proof. Let x0,y0 ∈ X, x0 = y0. If the M-topology is Hausdorff there are open set U,V such that x0 ∈ U,y0 ∈ V and U ∩ V = ∅. We may assume that U and V are intersections of finite collections of sets of the form (3). It follows that there is a set of the form (3) which contains x0 but not y0. I.e. x0 ∈ {y ∈ X : |l(y) − l(x)| < ǫ} while |l(y0) − l(x)| ≥ ǫ for some x ∈ X,l ∈ M and some ǫ > 0. Then l(x0) = l(y0), and we conclude that M separates the points of X. Klaus Thomsen 7. Consequences of the Hahn-Banach theorems
Weak vector space topologies Proof. Conversely, assume that M separates the points of X. Let x0,y0 ∈ X, x0 = y0. 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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