An introduction to the Hahn-Banach extension theorem: Part II
In this post we establish (in Lemma 2 below) the "key step" needed for the Hahn-Banach theorem (as described in the previous post), where we extend our linear functional by one real dimension. First we note an elementary fact about the operator norm for continuous linear functionals on real normed spaces. Lemma 1 (Operator norm for real linear functionals) Let $(E,\jnorm)$ be a normed space over $\R$, and let $\phi \in E^*.$ Then $(1)\Jdisplay\|\phi\|_\op = \sup \{\phi(x): x \in E, \|x\| \leq 1\,\}\,.$ Comment This looks very like the definition of the operator norm! However the subtle difference is that we work with $\phi(x)$ instead of $|\phi(x)|$ here. Our actual definition of the operator norm is that \[\|\phi\|_\op = \sup \{|\phi(x)|: x \in E, \|x\| \leq 1\,\}\,.\] Note that this would make no sense, as it stands, for complex normed spaces (though we could work with the real part of $\phi(x)$ in that case, and we would again obtain a true result). Let us refer to the ...