7. Consequences of the Hahn-Banach theorems - Aarhus Universitet
7. Consequences of the Hahn-Banach theorems - Aarhus Universitet
7. Consequences of the Hahn-Banach theorems - Aarhus Universitet
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Weak vector space topologies<br />
Pro<strong>of</strong>.<br />
Then S,T are X ′ -open and (x ′ ,y ′ ) ∈ S × T since li(x ′ ) ∈ Si and<br />
li(y ′ ) ∈ Ti,i = 1,2,... ,n.<br />
Fur<strong>the</strong>rmore, if s ∈ S,t ∈ T, we see that<br />
li(s + t) = li(s) + li(t) ∈ Si + Ti ⊆ Vi,<br />
proving that S + T ⊆ l −1<br />
1 (V1) ∩ l −2<br />
2 (V2) ∩ · · · ∩ l −1<br />
n (Vn) ⊆ U. I.e.<br />
S × T ⊆ R.<br />
The pro<strong>of</strong> that scalar multiplication is also continuous is left to <strong>the</strong><br />
reader. See ’Noter og kommentarer’.<br />
Klaus Thomsen <strong>7.</strong> <strong>Consequences</strong> <strong>of</strong> <strong>the</strong> <strong>Hahn</strong>-<strong>Banach</strong> <strong>the</strong>orems