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I am studying functional analysis by reading "Elements of Functional Analysis" by IJ Maddox (which was the set text for the Open University's now discontinued course on this subject).

In the Hahn-Banach theorem, what is the purpose of the 'dominating function'?

More specifically: we have vector space $V$, over the reals, and a subspace $U$. $f$ is a functional on $U$. There exists a function $p: U\rightarrow R$ such that $p(x+y) \le p(x) + p(y)$, $p(\lambda x)=\vert\lambda\vert p(x)$, and $f(x) \le p(x) $. The theorem states that $f$ can be extended to a functional on $V$.

My question is why $p$ is needed at all. The proof in Maddox's book does not appear to use $p$ except to verify that the extended function is still dominated by $p$.

Note that a similar question was asked on Aug 18 2013, by Chandu1729, but not directly answered.

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Jim Clow
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$p$ is not needed for the proof, but rather it's an important part of the formulation of the theorem. Not only can the linear functional be extended, but the extension is still dominated by $p$. Just an extension is not hard to produce, but the theorem actually tells you that you still have control over the extension knowing that the original functional is dominated. In applications of the theorem this is used often.

Ittay Weiss
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  • +1 for applications of controlling the extension. As an example, I had a problem recently that the solution given depended of that control. – Ivo Terek Jun 28 '15 at 20:42