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
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
<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 />
By Theorem 3.5 <strong>the</strong>re is a l ∈ X ∗ such that l(M0) = {0} and<br />
l(x0) = 1.<br />
Let x ∈ M. Then x − f (x)x0 ∈ M0 (since<br />
f (−f (x)x0) = f (x) − f (x)f (x0) = 0), and hence<br />
l(x − f (x)x0) = 0.<br />
It follows that l(x) = f (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