# Monthly Archives: December 2008

## Trigonometric Series and the Beginnings of Set Theory

Let $f\colon\mathbb{R}\to\mathbb{R}$ be a $2\pi$-periodic function. It may or may not have a representation as a trigonometric series

$\displaystyle{a_0+\sum_{n=1}^\infty a_n\sin(nx) + b_n\cos(nx)}$

A natural question to ask is whether or not the representation of $f$ as a trigonometric series is unique, if it has one. It was the consideration of this question that led Cantor to the invention of set theory.

There is a nice writeup of this story in the first part of this article by Alexander Kechris. I’ll give part of the story below.