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!)

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A related question on MathOverflow: mathoverflow.net/questions/26568/… – Jonas Meyer May 4 '11 at 20:07

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

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Some details on proof of the existence and some properties of Banach limits using Hahn-Banach Theorem are given in this answer: math.stackexchange.com/a/80571 – Martin Sleziak Jan 6 '12 at 13:59

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.

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Nice! Can I have a reference to this result ? – user50618 Jul 29 '14 at 13:58
@user50618 It is a famous result, you will find it up in Katznelson's book "An introduction to Harmonic Analysis" or Rudin's Functional Analysis. – AD. Jul 30 '14 at 18:18
@user50618 You have some references here. – Jean-Claude Arbaut Nov 12 '15 at 10:10

What i know about Hann Banach Theorem is the existence of enough functionals of a dual space on a given space, and these functionals sepearate points of the space. The sufficiency of these functional guaranteed enough maps in a dual space to work with.

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