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

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Weak vector space topologies Proof. Then S,T are X ′ -open and (x ′ ,y ′ ) ∈ S × T since li(x ′ ) ∈ Si and li(y ′ ) ∈ Ti,i = 1,2,... ,n. Furthermore, if s ∈ S,t ∈ T, we see that li(s + t) = li(s) + li(t) ∈ Si + Ti ⊆ Vi, proving that S + T ⊆ l −1 1 (V1) ∩ l −2 2 (V2) ∩ · · · ∩ l −1 n (Vn) ⊆ U. I.e. S × T ⊆ R. The proof that scalar multiplication is also continuous is left to the reader. See ’Noter og kommentarer’. 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: 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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