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 />
Theorem<br />
Theorem 3.10. Let X be a vector space and X ′ a separating vector<br />
space <strong>of</strong> linear functional on X. Then <strong>the</strong> X ′ -topology makes X a<br />
locally convex topological vector space whose dual is X ′ .<br />
The pro<strong>of</strong> is based on <strong>the</strong> following lemma <strong>of</strong> independent interest:<br />
Lemma<br />
Let l1,l2,... ,ln and l be linear functional on <strong>the</strong> vector space X.<br />
The following are equivalent:<br />
(a) There are scalars α1,α2,... ,αn such that<br />
l = α1l1 + α2l2 + · · · + αnln.<br />
(b) There is a γ > 0 such that |l(x)| ≤ γ max1≤i≤n |li(x)| for all<br />
x ∈ X.<br />
(c) For all x ∈ X, l1(x) = l2(x) = · · · = ln(x) = 0 ⇒ l(x) = 0.<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