I'm beginning with functional analysis and I have a related question. It concerns the Hahn-Banach theorem. In particular, I cannot appreciate its value, most probably because I don't understand it. I mean the Hahn-Banach theorem in it's most common form, like it is stated in the relevant Wikipedia article.
In particular, with the notation of the Wikipedia article, my (probably pretty dump) question is : why do we need the Hahn-Banach theorem to extend the bounded linear functional $\phi$ and don't we just consider its trivial extension, i.e., $\psi(x) = \phi(x), \forall x \in U$ and null everywhere else?
Best Answer
I think what many people have in mind is the following. Suppose $X$ is a normed vector space, $Y$ is a subspace of $X$, and $f$ is a bounded linear functional on $Y$. Then $Y$ has an algebraic supplement $Z$, that is $X=Y+Z$. (Of course, the supplement may not be topological. Hence, the "$+$" sign.) We want to define $f$ on $X$ such that $f(x)=f(y+z)=f(y)$, that is to require $f$ maps $Z$ to $0$. This is definitely linear, but may not be bounded in infinite dimensional spaces. What could happen is $\|y+z\|$ may be small, but $f(y+z)=f(y)$ may be large. This is in fact the key point in the proof of the Hahn Banach Theorem (the extension one): In extending $f|_Y$ to $f|_{Y+\mathbb{R}\cdot z}$, the selection of $f(z)$ is crucial.