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

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Weak vector space topologies Proof. In order to conclude that τ ′ makes X into a topological vector space it remains now only to show that every one-point set {x} in X is closed. This goes as follows: For each y = x there is an element ly ∈ X ′ such that ly(x) = ly(y). This is because X ′ separates points by assumption. It follows that {x} c = {z ∈ X : ly(z) = ly(x)} . y∈{x} c Klaus Thomsen 7. Consequences of the Hahn-Banach theorems
Weak vector space topologies Proof. In order to conclude that τ ′ makes X into a topological vector space it remains now only to show that every one-point set {x} in X is closed. This goes as follows: For each y = x there is an element ly ∈ X ′ such that ly(x) = ly(y). This is because X ′ separates points by assumption. It follows that {x} c = {z ∈ X : ly(z) = ly(x)} . Since y∈{x} c {z ∈ X : ly(z) = ly(x)} = l −1 y (Φ\{ly(x)}) , we see that {x} c is X ′ -open. 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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