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 />
(c) ⇒ (a): Define π : X → Φ n such that<br />
π(x) = (l1(x),l2(x),... ,ln(x)).<br />
It follows from (c) that we can define l0 : π(X) → Φ such that<br />
l0 (π(x)) = l(x).<br />
Let P : Φ n → π(X) be a linear projection such that P (Φ n ) = π(X).<br />
Then F = l0 ◦ P : Φ n → Φ is a linear map such that<br />
F ◦ π(x) = l0 ◦ P ◦ π(x) = l0 (π(x)) = l(x).<br />
Recall from linear algebra that <strong>the</strong> linear map F : Φ n → Φ is <strong>of</strong> <strong>the</strong><br />
form<br />
F(x1,x2,... ,xn) = α1x1 + α2x2 + · · · + αnxn.<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