Applications of the Hahn-Banach Theorems

Question: What are some interesting or useful applications of the Hahn-Banach theorem(s)?

Motivation: Most of the time, I dislike most of Analysis. During a final examination, a question sparked my interest in the Hahn-Banach theorem(s). One of my favorite things to do is to write a math blog (mlog?) post about various topics so that I can better understand them, but I know very little about Hahn-Banach and a quick google search didn't seem to point to anything neat. I was interested in seeing what you all liked (if anything!) about the Hahn-Banach Theorems.

Also, I can't seem to make this a community wiki, but I think it ought to be one. If someone could either fix this, I would appreciate it! (If not, please delete this!)

How about the Wiener Tauberian theorem:

Theorem (N. Wiener 1932). For $f\in L^1(\mathbb{R})$, let $X= \operatorname{span}\{f_t:t\in\mathbb{R}\}$ (that is the linear subspace spanned by the translates of $f$). Then the closure of $X$ in $L^1$ is $L^1$ if and only if the Fourier transform of $f$ has no zero.

Which, in itself, has applications in many different fields running from number theory to PDE.

One I know of is the hyperplane separation theorem for convex sets. Another is the existence of Banach generalized limits.