This inspired this post: Is there some theorem that you were initially impressed with but are now far less impressed? I list things I have heard of for this category. I request that you submit your own examples.
- Every number is the sum of 4 squares. This is impressive and still is. Number theorist must use this all the time! Alas, aside from its use in Hilbert's 10th problem, and maybe a few few other places, I has never seen it used and is now less impressed. However, this may be unwarranted. Some theorems in math are impressive for the achievement, others for their use later. This one IS impressive for its achievement. But, as far as I can tell (and I could be wrong), not for its uses.
- Every group is a group of permutations. Group Theorists must use this all the time! Alas the proof makes you realize its more of a tautology. Rarely used by Group Theorists. It is used in some of the proofs of Sylow's theorem. I do not know of any others uses. And this one is not impressive for its achievement.
- The Prime Number Theorem. Since results that are very very close to it can be gotten with much much much less effort, getting the actual constant down to 1 seems like too much sugar for a cent. (For more on PNT and a link to an easy proof of a weaker version see an old post of mine here.) However, this one is an achievement certainly. And it inspired other great mathematics.
- Poincare's Conjecture says that if something looks, feels, and smells like a sphere, then its a sphere. Is that really worth $1,000,000? Perhaps Perelman didn't think so either.