10:35 PM

So, continuing my discussion of the big picture of real analysis, I ended last time talking about sequences of functions, and the idea that under the right conditions (the sequence is uniformly convergent, or the sequences of derivatives is uniformly convergent), the members of the sequence of functions have the same properties as the function which is the limit of the sequence. This means that if you have a function which is computationally ugly, you can potentially rewrite it as a sequence of functions which are easy to work with.

There are two important examples of sequences of functions which converge uniformly. The first is polynomials. Polynomials are easy to work with, so if you can rewrite an ugly function as a polynomial, you can turn hard problems into easy problems. In particular, Taylor series are a particular sequence of polynomials which approximate a function. Taylor series have important limitations, in that the original function must be infinitely differentiable and Taylor series do not always converge, but when they do work, they are a powerful and convenient tool.

The second example of a sequence of functions which converges uniformly is the Fourier series. Given a function which is bounded and periodic (meaning it repeats itself), you can write the function as a sum of sine and cosine functions. Sines and cosines are difficult to evaluate, but they are easy to work with (for example differentiation and integration). Their smoothly undulating curves are also pretty. It's aesthetically appealing to be able to convert an angular, sharp edged function into a sum of beautiful waves.

I've previously encountered Fourier series in other contexts, and while I studied the basic math to some extent, there was an element of, "we are justified in using Fourier series because they give the correct results in practice." I found it personally satisfying to come back to them and be able to say that we are mathematically justified in using them because we can mathematically prove that they give the expected results.

This is as far as I've gone in studying real analysis. It all comes back to using limits as a tool to say that these are the conditions under which we are allowed to do certain mathematical operations, and these are the conditions under which the operations will fail. Along the way, I started with the concept of sequences of numbers, and eventually extended that idea to sequences of functions. The fact that functions can be inserted in a place where I expected to use numbers has also had the effect of changing how I think about functions in general.

Real analysis goes on from here, leading to questions like "what is the mathematical definition of length?" (Think about it. A line segment has some length. But a line is made up of points, and points have no length. So where does the length of the line come from?) This leads to questions like, "can you have a set of points which is not a line but which also has a length?" I'm interested in these questions, which start to have a metaphysical significance, but I'm happy to stop here with my current studies for now.

0 comments