TITLE: Why is the Hahn-Banach theorem so important?
QUESTION [56 upvotes]: Every time I hear it mentioned it is praised in the highest possible terms, and I remember one of my old lecturers saying that it is one of the 3 most important theorems in analysis. Yet the only consequences of it that I have read is that it proves that there are lots of functionals and that separating hyperplanes exist. Are those 2 consequences really that spectacular, or are there other ones that I don't know of?

REPLY [46 votes]: I have used it (sometimes with coauthors) several times in the following general context. I have wanted to prove that a function f can be decomposed as a sum g+h, where g has certain properties and h has certain properties. It has been possible to show that the set of acceptable g is convex, as is the set of acceptable h. So I'm trying to show that f belongs to a sum of two convex sets K+L. But a sum of convex sets is convex, so if f cannot be written in such a way, then it can be separated from K+L by a functional. It is often possible to derive a contradiction from this.
The result is that one can prove the existence of the desired decomposition under circumstances where explicitly defining a decomposition would be difficult.
Incidentally, the applications I am alluding to are of the finite-dimensional Hahn-Banach theorem. Some people call it the minimax theorem, and still others would call it duality for linear programming or something like that.