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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<strong>Consequences</strong> <strong>of</strong> <strong>the</strong> <strong>Hahn</strong>-<strong>Banach</strong> <strong>the</strong>orems<br />
Theorem<br />
(The corollary on p. 59) Let X be a locally convex topological<br />
vector space. Then X ∗ separates <strong>the</strong> points <strong>of</strong> X, i.e. when x = y<br />
in X <strong>the</strong>re is a l ∈ X ∗ such that l(x) = l(y).<br />
Pro<strong>of</strong>.<br />
Apply <strong>the</strong> <strong>Hahn</strong>-<strong>Banach</strong> separation <strong>the</strong>orem, Theorem 3.4 b), with<br />
A = {x} and B = {y}.<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