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 />
(Theorem 3.5) Let X be a locally convex topological vector space<br />
and M ⊆ X a subspace. If x0 ∈ X is NOT in <strong>the</strong> closure M <strong>the</strong>re is<br />
a functional l ∈ X ∗ such that l(M) = {0} and l (x0) = 1.<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 = M and B = {x0} to obtain L ∈ X ∗ and γ ∈ R such that<br />
Re L(M) < γ < Re L(x0).<br />
Since 0 ∈ M we conclude that γ > 0. Set l = L(x0) −1 L.<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