TITLE: About Hahn Banach, can I get an example problem outside of functional analysis?
QUESTION [12 upvotes]: I just started reading functional analysis and came across the Hahn-Banach theorem. Everywhere I looked stated this was very important and it is because it allows us to enlarge a dual space $X^+.$ I keep wondering to myself, so what? (sorry for sounding rude). If we can enlarge the dual space of $X^+$, then what? Why are the extensions of linear functionals so important? I mean for a subspace $Z \subset X$, is there something important we need to know about $X-Z$ that we need to extend a linear functional $f: Z \to \mathbb{F}$ on?
Is there an application or simple example of this outside of functional analysis that motivates the idea? Even something simple in Differential Equations would be insightful or a simple problem in Linear Algebra (actually this is probably dumb since I would imagine you would just extend the basis, so we have to give an example for a infinite dim space; maybe showing an analogue of Hahn-Banach in this setting would be neat and somewhat convincing)
Right now I just can't be bothered reading the full proof of something that I don't see the importance in and only to forget about it later...
Thank you.

REPLY [4 votes]: I personally really like the following application to a very natural and basic question in measure theory:
There exists a finitely additive measure $m:\mathcal{P}(\mathbb{R}) \to \mathbb{R}_{\geq 0}$ defined on all subsets of the non-negative real line that are translation invariant (i.e. $m(x+A) = m(A)$) that agree with the usual Lebesgue measure on measurable sets. The proof relies on the Hahn Banach theorem and is done in an ingenious way, one can find it in Stein and Shakarchi's book on functional analysis.