The Fourier Transform can look like a complicated integral involving complex exponentials.

But underneath it, there is a surprisingly simple geometric idea: the inner product.

In this video, we build the Fourier Transform from ideas you already know.

We start with ordinary vectors and dot products, extend the idea of an inner product to functions, understand what it means for two functions to be orthogonal, and finally see complex exponentials as directions in a function space.

From there, the Fourier Transform becomes much more intuitive: it measures how much of each frequency is present in a function.

0:00 Introduction to the topic
1:37 Historical context
4:47 Vector spaces and basis
9:52 Extracting information
17:09 Generalizing vector spaces
21:33 Inner products of functions
27:39 Complex exponentials
35:16 Defining Fourier transform