Before I got distracted by complex numbers, I had been talking about euclidean spaces. I left off after introducing the norm, or length, of a vector in euclidean space. As a continuation of that topic, I want to look at the properties of norms in euclidean spaces. Specifically, if x and y are vectors in a euclidean space, let's look at ||x−y||, or the distance from y to x.
||x−y|| is always greater than or equal to 0. In fact, it's only equal to 0 if x=y. Since x−y is a vector, it is clear that the distance is always positive because the norm is calculated as a square root, which is a positive number.
||x−y|| = ||y−x||. The norm is a square root of a dot product of a vector with itself. The dot product is a sum of products, and since both vectors are the same, it becomes a sum of squares. Since a2=(−a)2, the order of the subtraction of x and y does not matter.
||x−y||≤||x−z||+||z−y||. This follows from the triangle inequality, as mentioned in my previous posts about euclidean spaces.
These three properties were written in terms of norms, but they could be rewritten as distances. And just like fields as a general concept were defined from the properties of addition and multiplication for the rational numbers, we can define metric spaces based on these three properties for distance. These three properties define a distance function, and any set of points with a defined distance function that has these properties for every point in the set is a metric space.
This is where I get nervous. I've studied abstract algebra enough to have some familiarity with fields other than the real numbers or complex numbers, so I have some sense of the distinction between the two, and I feel like I can recognize if something is true for the real numbers but might not be for fields in general. In addition, my impression so far is that real analysis isn't dependent on fields as much as it is on the real numbers, so confusion on the differences is not likely to be a problem for studying real analysis.
I have not studied topology, and metric spaces are (I think) a topology concept. It's also not clear to me how dependent real analysis is on metric spaces. I am somewhat concerned that I will take my one example of metric spaces (Euclidean k-spaces with distance defined as the square root of the dot product) and I will go off and draw all sorts of invalid conclusions about metric spaces in general. But this is what the textbook and course notes are giving me to work with, so I will press on and hope against catastrophic errors.