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 />
Let X be a real or complex vector space, and M a collection <strong>of</strong><br />
lineat functionals X → Φ.<br />
The sets <strong>of</strong> <strong>the</strong> form<br />
{y ∈ X : |l(y) − l(x)| < ǫ} (3)<br />
where x ∈ X,ǫ > 0 and l ∈ M vary, is a subbase for a topology on<br />
X,<br />
namely <strong>the</strong> topology where a subset <strong>of</strong> X is open if and only if it is<br />
<strong>the</strong> union <strong>of</strong> sets which are <strong>the</strong> intersection <strong>of</strong> a finite collection <strong>of</strong><br />
such sets.<br />
This will be called <strong>the</strong> M-topology <strong>of</strong> 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