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 />
Conversely, assume that M separates <strong>the</strong> points <strong>of</strong> X. Let<br />
x0,y0 ∈ X, x0 = y0.<br />
There is <strong>the</strong>n a functional l ∈ M such that l(x0) = l(y0).<br />
Set ǫ = 1/2 |l(x0) − l(y0)| > 0, and note that<br />
Since<br />
x0 ∈ {y ∈ X : |l(y) − l(x0)| < ǫ} ,<br />
y0 ∈ {y ∈ X : |l(y) − l(y0)| < ǫ} .<br />
{y ∈ X : |l(y) − l(y0)| < ǫ} ∩ {y ∈ X : |l(y) − l(x0)| < ǫ} = ∅,<br />
<strong>the</strong> pro<strong>of</strong> is complete.<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