In this course, the tools of linear algebra start to be applied in a realm where vectors (finite sequences of numbers) are replaced by continuous functions. This may seem very odd, at first, but it makes sense if you think of a continuous function as an infinite sequence of numbers, indexed not by positions (1,2,3), but by the real numbers themselves. One coordinate per real number!

Much as vectors can be decomposed in various ways as the sum of simpler functions, (remember Gram-Schmidt?), functions can be decomposed and written as a sum of "parts" in several very interesting ways. First, a (continuous periodic) function can be written as a sum of simple sinusoidal functions, giving the Fourier decomposition. If the original function is played on good quality speaker device, the simple sinusoidal parts will be the pitches distinguished by the ear, and their contribution to the sum will correspond to their relative perceived volumes.

Functions can also be decomposed as the sum of polynomial functions (The Stone- Weierstrass theorem). This idea is strongly related to the notion of the Taylor series.

In order to prove these theorems, you'll have to learn more sophisticated integration theory. In particular, you'll learn to replace the Riemann integral with the more general Lebesgue integral. You will also learn about measure theory, which is the rather difficult answer to a simple question: "Given any arbitrary subset of the real numbers, how do we assign it a length?" If this question seems simplistic, let me convince you otherwise: How much of the total length of the integral [0,2] should be attributed to the rational numbers in that range, and how much of the length belongs to the irrationals?