Deep theorem with trivial proof [closed]
It is the snobbishness of the young to suppose that a theorem is trivial because the proof is trivial.
-- Henry Whitehead
I have been awestruck by the beauty of this quote.
What is in your opinion a good contender to exemplefy the meaning intended by Whitehead?
Off the top of my head I am thinking of Langrange's theorem in Group Theory, which is rather simple to prove but provides a very useful insight.
Cantor's diagonal argument shows not only the completely unintuitive result that uncountable sets exist, but that something as everyday as the real numbers belongs to that category.
The proof may be explained even to those with no mathematical background by a simple drawing.
Stokes' Theorem is a "deep theorem" with a trivial proof. Here is a quote from Spivak's Calculus on Manifolds:
Stokes' theorem shares three important attributes with many fully evolved major theorems:
- It is trivial.
- It is trivial because the terms appearing in it have been properly defined.
- It has significant consequences.
Since this entire chapter was little more than a series of definitions which made the statement and proof of Stokes' theorem possible, the reader should be willing to grant the first two of these attributes to Stokes' theorem. The rest of the book is devoted to justifying the third.
Stokes' Theorem includes as a special case:
1. The fundamental theorem of calculus.
2. Green's Theorem.
3. The divergence theorem (from multivariable calculus).
On top of all that, the statement is as elegant as it gets: $$\int_M\partial\omega=\int_{\partial M}\omega$$
This is going to depend on the definitions of deep and trivial, but the following just might qualify:
Every map defined on some basis of a vector space admits a unique linear extension to the whole vector space. What's more, the map is injective/surjective/bijective if and only if it maps the basis to a linearly independent system/ spanning system/basis.
Indeed the proof writes itself straight from the basic definitions, yet the statemente is, in a sense, the essence of linear algebra.