I want to write about something very cool I learned about in my Linear Algebra class the other day.
Since since the sum of two continuous functions is continuous, and a continuous function multiplied by a scalar is continuous, the set of all continuous functions defined on the domain [a,b] is a vector space.
You can define an inner product [F|G] as the integral of the product F*G from point a to b (This operation has everything it takes to be an inner product). The inner product naturally gives us a norm. ||F|| = sqrt([F|F]), and an angle between functions: [F|G] = cos(theta) * ||F||||G||. The norm gives us a distance ||F-G||.
So now we have distance between functions and angles between functions. That means we can project a function onto another the same way that we can project a vector onto another in R^n. That’s awesome! And potentially useful. =P
(Edit: Yes, it’s useful. You can deduce the Fourier Series by noting that the cosine and sine as basis for this vector space)