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 />
X is locally convex: Consider an X ′ -open set U such that 0 ∈ U.<br />
There are <strong>the</strong>n pairs (li,Vi),i = 1,2,... ,n, such that<br />
0 ∈ l −1<br />
1 (V1) ∩ l −2<br />
2 (V2) ∩ · · · ∩ l −1<br />
n (Vn) ⊆ U,<br />
where li ∈ X ′ and <strong>the</strong> Vi’s are open in Φ.<br />
Note that 0 = li(0) ∈ Vi for all i. Since Φ is a locally convex<br />
topological space, <strong>the</strong>re are open and convex neighborhoods Wi <strong>of</strong><br />
0 in Φ such that Wi ⊆ Vi.<br />
Since <strong>the</strong> li’s are linear, l −1<br />
1 (W1) ∩ l −2<br />
2 (W2) ∩ · · · ∩ l −1<br />
n (Wn) is<br />
convex.<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