## Convergence in metric spaces

For many mathematics majors, the first-semester elementary analysis course serves as an introduction to formal mathematics. The central concept in this course is, not surprisingly, that of a “limit”, and what “convergence” really means. While high school calculus courses may have dealt with this through heuristic reasoning, this is suddenly no longer sufficient and we instead concern ourselves with long definitions, often chock-full of wonderful quantifiers.

I’d like to remark that in such a course, these concepts, in spite of their breathtaking generality, are introduced in the setting of the familiar real number system. I am not, however, dismissing the importance of $\mathbb{R}$ since, being the one and only complete ordered field, it is undeniably a rather nice structure to work with. Especially when discussing distance, for after all, what is a distance if not some real number?

Defining this notion of the distance between elements on sets that consist of elements other than numbers, as it turns out, is often useful. For example, it seems perfectly reasonable to consider a sequence of functions, defined on some domain, and ask whether perhaps this sequence converges to some limiting function. Note that all we need to talk about limits is, in effect, this notion of distance. So the concept of what is called a metric space arises naturally.

Definition. A metric space $(X,d)$ is simply any set $X$ together with a function, $d : X \times X \to \mathbb{R}$, called the distance function or metric, that maps pairs of elements in $X$ to real numbers, and satisfies the following properties where $x, y, z$ are elements of $X$:

1. $d(x,y) \geq 0$, holding with equality if and only if $x = x$
2. $d(x,y) = d(y,x)$
3. $d(x,z) \leq d(x,y) + d(y,z)$

Note that these properties agree with our intuitive notion of what distance means. The first property says that the distance function’s value, for any pair of elements, is nonnegative. The second property says that the distance from an element $x$ to an element $y$ is the same as the distance from the element $y$ to the element $x$. That is, the distance function is commutative. Thirdly, we have a very important property called the triangle inequality, which says that the distance from one point directly to another point must be less than or equal to the distance between those points following any path that includes an intermediate point. This last property is what really disqualifies many functions from being metrics, and also appeals to our intuition: the shortest route is the direct route.

With this laid out, we can use metric spaces to formally define what it means for a sequence of functions to converge to a limit. In elementary analysis, convergence is often defined by talking about absolute values. As you probably guessed, the absolute value $|x-y|$ for $x, y \in \mathbb{R}$ is an example of a metric on $\mathbb{R}$.

Theorem. Let $(X, d)$ and $(Y, d')$ be metric spaces. We denote by $\mathcal{C}^0(X, Y)$ the set of all continuous, bounded functions $f : X \to Y$. Let us define for all $f, g \in \mathcal{C}^0(X,Y)$ the function $\rho(f,g) = \sup \{ d'(f(x), g(x)) \mid x \in X \}$. (The supremum exists, because the functions in question are bounded.) We claim that $(\mathcal{C}^0(X,Y), \rho)$ is a metric space.
Proof. By using the fact that $d'$ is a metric on $Y$, you can verify that $\rho$ indeed constitutes a metric for the set of functions $\mathcal{C}^0(X,Y)$, and hence that $\mathcal{C}^0(X,Y)$ is a metric space.

Theorem. If $X$ and $Y$ are complete metric spaces, then $\mathcal{C}^0(X,Y)$ is also a complete metric space.

We now define a limit in this new context of a metric space.

Definition. We say that a sequence $(x_n)_{n=1}^\infty$ in a metric space $(X,d)$ converges to a point $x_0 \in X$ if for every $\epsilon > 0$ there exists some $N \in \mathbb{N}$ such that for all $n \geq N$ we obtain $d(x_n, x_0) < \epsilon$.

(I’ll keep writing on this later.)