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 />
Pro<strong>of</strong> <strong>of</strong> Theorem 3.10. Almost by definition, <strong>the</strong> open sets in<br />
X ′ -topology are <strong>the</strong> sets that are unions <strong>of</strong> sets <strong>of</strong> <strong>the</strong> form<br />
l −1<br />
1 (V1) ∩ l −2<br />
2 (V2) ∩ · · · ∩ l −1<br />
n (Vn),<br />
where each li ∈ X ′ and Vi is an open subset <strong>of</strong> <strong>the</strong> scalars Φ.<br />
To show that X is a topological vector space in <strong>the</strong> X ′ -topology we<br />
check first that <strong>the</strong> vector space operations are continuous.<br />
To show that addition is continuous, let U be an X ′ -open set. We<br />
must show that<br />
is X ′ -open in X × X.<br />
R = {(x,y) ∈ X × X : x + y ∈ U} (4)<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