I learned from E. Kowalski’s blog a nice theorem of Kocher and Specker:
There does not exist a function , such that for any which are orthogonal to each other, .
The statement here was at first counter-intuitive, so I tried to prove it. I found some kind of geometrical proof, I think it’s nice (and I hope it’s right!):
Proof: Label all by and identify points of with vectors of . For every othogonal triple one and only one of the labels is . Therefore, for every “zero” label (call it “north pole”), the “equator” will be made of “one”s, and the “south pole” will be a “zero”.
Consider labels “zero” which have an angle between them , there are two intersecting (non-orthogonal!) equators of “one”s. Now while a point moves on one of the equators, there is exactly one point on the other equator which is orthogonal to it: as describes the first equator, will therefore describe the second equator. The third point completing an orthogonal basis , will also describe a curve of “zero”s: it is a circumference having the two “zero”s as diameter!
Iterating the above construction it is easy to see that the set of zeroes is open. But since the set of “one”s depends continuously on the set of “zero”s (through the “equator-pole” construction), it is also open! Contraddiction (since the sphere is connected).