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 3.6) Let X be a locally convex topological vector space<br />
and M ⊆ X a subspace. If f : M → Φ is a continuous linear<br />
functional on M, <strong>the</strong>re is an l ∈ X ∗ such that l|M = f .<br />
Pro<strong>of</strong>.<br />
We may assume that f is not identically 0.<br />
Set<br />
M0 = {x ∈ M : f (x) = 0}.<br />
and choose x0 ∈ M such that f (x0) = 1.<br />
Then x0 /∈ M0: See <strong>the</strong> next slide!<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