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 />
(<strong>of</strong> lemma): (a) ⇒ (b):<br />
Set γ = n max1≤i≤n |αi|. Then<br />
|l(x)| ≤ |α1| |l1(x)| + |α2| |l2(x)| + · · · + |αn| |ln(x)|<br />
≤ n max<br />
1≤i≤n |αi| max<br />
1≤i≤n |li(x)| = γ max<br />
1≤i≤n |li(x)| .<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