Metric Spaces I (Overview) Saturday, August 31, 2024

Metric spaces, are in essence, the realm of analysis (in my experience). Analysis (or as I have heard from those my senior, “applied triangle inequality”) relies on notions of distance and, in particular, arbitrarily precise approximations. (e.g., the $ \varepsilon $-$ \delta $ arguments that you might have seen previously.)

Metric spaces and examples thereof

Consider three different locations A, B, C in our world. Experience tells us that

1. To get from point A to point A should be $ 0 $ since we don’t have to move at all.
2. The distance from point A to point B should be the same as the distance from point B to point A.
3. Walking from A to B to C takes longer than walking from A to C directly.

From this, we get three basic rules that any “sensible” sense of distance should follow: if $ X $ is a space (read: set), a distance function $ d \colon X \times X \to \mathbb{R^{\geq 0}} $ satisfies for all $ x,y,z \in X $:

1. $ d(x,x) = 0 $,
2. $ d(x,y) = d(y,x) $, and
3. $ d(x,y) + d(y,z) \geq d(x,z) $.

Any function satisfying this description is henceforth known as a metric on $ X $.

Here are many examples of metric spaces:

1. $ X = \mathbb{R} $ and $ d(x,y) = |x-y| $
2. $ X = \mathbb{R}^n $ and $ d(x,y) = [\sum_{i=1}^{n} (x_i-y_i)^2]^{1/2} $
3. $ X = \mathbb{R}^n $ and $ d(x,y) =\sum_{i=1}^{n} |x_i-y_i| $
4. $ X = \mathbb{R}^n $ and $ d(x,y) = \sup_{i=1}^n |x_i-y_i| $

Notice that we can have different metrics on the same underlying set. Neat! But now we have a few slightly more exotic examples:

5. $ X = C([0,1], \mathbb{R}) \coloneqq \left \{ f \colon [0,1] \to \mathbb{R} \mid f \text{ continuous} \right \} $ and $ d(f,g) = \sup_{x \in [0,1]} |f(x)-g(x)| $
6. If $ (X,d) $ is a metric space and $ F \subseteq X $, then $ (F,d) $ is a metric space
7. Let $ X $ be any set and take $$\begin{equation} d(x,y) = \begin{cases} 1 \quad & x \neq y\\\ 0 \quad & x = y \end{cases} \end{equation}$$

Sequences

We now delve into the one of the most fundamental objects of study in analysis: the sequence.

A sequence is a list of elements from a set (like numbers from $ \mathbb{R} $) indexed by the positive integers $ \mathbb{Z}^+ $.¹

To put say this more precisely, given a set $ X $, a sequence is a function $ p \colon \mathbb{Z}^+ \to X $. We also often denote $ p(n) $ by $ p_n $ for simplicity, and we write $ (p_n) $, $ (p_n)_n $, $ (p_n)_{n=1}^\infty $ for the sequence itself.

We also have an object called the subsequence, literally a sequence from within a sequence. A subsequence of a sequence $ (p_n) $ is sequence $ (q_m) $ defined by $ q_m = p_{\iota(m)} $ where $ \iota \colon \mathbb{Z}^+ \to \mathbb{Z}^+ $ is a function such that $ \iota(1) < \iota(2) < \iota(3) < \cdots $. We often refer to $ \iota(m) $ as $ n_m $. A shorthand for a subsequence is then $ (p_{n_m})_m $.

From our definition of subsequence, we have an immediate consequence: for all $ m \in \mathbb{Z}^+ $, $ \iota(m) = n_m \geq m $.

An understanding of long term behavior

The power of sequences is not necessarily in simply what occupies your list; it is moreso in its ability to represent the long term behavior of an object.

In particular, these infinitely-long lists can approach or converge to something. There’s obviously very simple cases like the sequence $ p_n = 0 $ for all $ n $, but so much can be studied in this paradigm. However, things can also not converge, like $ p_n = (-1)^n $ or $ p_n = n $.

From this idea of long term behavior, two big ideas fall out: convergence (approaching a single point, or none) and Cauchy-ness (how close the terms get to one another).

We now define both of these a bit more rigorously. Given a metric space $ (X,d) $, we say a sequence $ (p_n) $ in $ X $ converges to $ p \in X $ if for all $ \varepsilon > 0 $ there exists an $ N \in \mathbb{Z}^+ $ such that $ n \geq N $ implies that $ d(p_n,p) < \varepsilon $. In other words $$\begin{equation} (\forall \varepsilon > 0)(\exists N \in \mathbb{Z}^+)[n \geq N \implies d(p_n,p) < \varepsilon ]. \end{equation}$$ We denote this by $\lim_{n\to\infty} p_n = p $ or $ p_n \to p $.

In other other words, our sequence converges to $ p $ if we can get arbitrarily close to $ p $. Now we say our sequence is Cauchy if for all $ \varepsilon > 0 $ there exists $ N \in \mathbb{Z}^+ $ such that whenever $ n,m \geq N $, $ d(p_n,p_m) < \varepsilon $. In quite similar words as before, $$\begin{equation} (\forall \varepsilon > 0)(\exists N \in \mathbb{Z}^+)[n,m \geq N \implies d(p_n,p_m) < \varepsilon]. \end{equation}$$ In other other quite similar words as before, our sequence is Cauchy if the terms of the sequence get arbitrarily close together.

Boundedness

We first have to discuss sets being bounded. A set is bounded if it can be entirely contained in a ball of finite radius around some point of the space. Now, a sequence $ (p_n) $ is bounded if the set $ \left \{ p_n : n \in \mathbb{Z}^+ \right \} $ is bounded. Symbolically, $ (p_n) $ is bounded if and only if: $$\begin{equation} (\exists q \in X)(\exists r > 0)\left [ \left \{ p_n : n \in \mathbb{Z}^+ \right \} \subseteq \left \{ p \in X : d(p,q) < r \right \} \right ] \end{equation}$$

Some results on sequences

For the following proofs, let $ (X,d) $ be a metric space with a sequence $ (p_n) $ in $ X $.

Statement #1.

Statement #2.

Statement #3.

Statement #4.

Statement #5.

Statement #6.