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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<strong>Consequences</strong> <strong>of</strong> <strong>the</strong> <strong>Hahn</strong>-<strong>Banach</strong> <strong>the</strong>orems<br />
Pro<strong>of</strong>.<br />
Remember that M is equipped with <strong>the</strong> relative topology inherited<br />
from X. Since f : M → Φ is continuous with respect to this<br />
topology, M0 = f −1 ({0}) is closed in that topology.<br />
This means that M\M0 = M ∩ W for some open set W <strong>of</strong> X.<br />
Then<br />
W ∩ M0 = W ∩ M ∩ M0 = (M\M0) ∩ M0 = ∅.<br />
This means that M0 ⊆ W c and it follows that M0 ⊆ W c .<br />
Since x0 ∈ M\M0 = M ∩ W, we conclude that x0 /∈ M, as asserted.<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