While I was driving in my car today, I thought of a proof of the Cauchy-Schwartz theorem. I’m sure that it is completely unoriginal, but it has the advantages of both being longer and requiring more background than the usual proof (which you can find on the Wikipedia page).

Let (,) be an inner product. From the definition, we know that for two vectors x, y and two scalars a, b that

(ax+by, ax+by) = a² (x,x) + 2ab (x,y) + b² (y,y) ≥ 0

This is a positive-definite quadratic form in a, b, which means that its associated matrix has positive determinant:

(x,x) (y,y) – (x,y) (x,y) ≥ 0,

which is the result.

The real advantage of the proof, I suppose, is that if you already have the linear algebra background there’s no trick involved. It also means that using the same determinant argument there are analogues of the inequality that involve n vectors instead of two.

I’m intrigued by the beginning of a new series of posts at the Everything Seminar about harmonic analysis. This particular post talks about the relationship of singular integral operators and Carleson’s Theorem. Carleson’s Theorem (that Fourier series of functions in L^p for p > 1 converge pointwise almost everywhere) is a famously difficult result; the post gives some idea of where the difficulty lies.

For the ambitious, a complete proof is available in a preprint by Michael Lacey.

Calculus has been the subject of immense amounts of educational material, ranging from textbooks to blog posts. Unfortunately, that is now all obsolete. The definitive presentation of calculus is here: Calculus, the Musical.

Charles at Rigorous Trivialities has written a post outlining the proof of two pretty theorems from the invariant theory of finite groups: Noether’s theorem that the ring of invariants is finitely generated, and Molien’s formula for the number of homogeneous invariants of a given degree.

Not only is John Armstrong a failed crackpot, he is wrong about statistics. Statistics is, from the mathematical point of view, a perfectly interesting subject; this fact is carefully concealed from us by statisticians. For example, most mathematicians know the central limit theorem, which says that the sum of large numbers of independent, identically distributed (iid) random variables tend to be normally distributed. This even has an elegant proof in terms of Fourier analysis, where addition of random variables because multiplication of Fourier transforms.

What mathematicians don’t know is that almost every other statistic ever defined also satisfies the central limit theorem. The median of a large number of iid random variables? Normally distributed. The mode of a large number of iid random variables (where the underlying distribution has a single mode)? Normally distributed. The cosine of the seventeenth percentile? Normally distributed. The simplest explanation for this cavalcade of normality involves the Gâteaux derivative in functional analysis.