1.4 Metric Spaces (Non-examinable)

Have you noticed that both the absolute value on the real numbers and the modulus on the complex numbers behave suspiciously alike? They even have the same notation! This is because they’re both examples of metrics on a set:

Definition 1.4: (Metric Space)

A metric space \((X,d)\) consists of a set \(X\) together with a function \(d: X\times X \to \mathbb{R}\) satisfying \(\forall x,y,z \in X:\)

  1. Non-negativity: \(d(x,y) \geq 0,\)
  2. \(d(x,y) = 0 \; \Leftrightarrow x = y,\)
  3. Symmetry: \(d(x,y) = d(y,x),\)
  4. Triangle Inequality: \(d(x,z) \leq d(x,y) + d(y,z).\)

As we’ve seen, \((\mathbb{R}, \lvert \cdot \rvert)\) and \((\mathbb{C}, \lvert \cdot \rvert)\) both define metric spaces. The existence of this definition suggests that we can define the distance between two members of a set in different ways! For example, thinking about the complex numbers \(X = \mathbb{C},\) with complex numbers \(z_1 = a_1 + ib_1\) and \(z_2 = a_2 + ib_2,\) we could instead look at:

  • The discrete metric \[d(z_1,z_2) = \begin{cases} 0 \quad \text{if $z_1=z_2$},\\ 1 \quad \text{otherwise}\end{cases}\]
  • The \(p\)-norm: \[d(z_1,z_2) = \left(\lvert a_1-a_2\rvert^p + \lvert b_1-b_2\rvert ^p\right)^{1/p},\;\; 1\leq p < \infty.\]
  • The \(\infty\)-norm: \[d(z_1,z_2) = \max\left\lbrace \lvert a_1 - a_2 \rvert, \lvert b_1 - b_2 \rvert\right\rbrace.\]

These last two are called norms because they satisfy some additional properties, but don’t worry about these until next year! It’s an interesting thing to look at what the unit circle looks like under the \(1\)-norm (blue), \(2\)-norm (red) and \(\infty\)-norm (green). The complex numbers used here are \(z_1 = x + iy\) and \(z_2\)=0:

We can even look at defining distances on other sets. For example, consider the set \(X = C^{0}([a,b])\), which is the set of all continuous functions \(f:[a,b] \to \mathbb{R}.\)3 One way we can define the distance between two functions \(f\) and \(g\) is to consider evaluating them at all members of the domain, and finding what the maximum deviation between the two are. This gives rise to a metric \(d:X\times X \to \mathbb{R},\) namely \[d(f,g) := \max_{x \in [a,b]} \lvert f(x) - g(x) \rvert.\] Have a go at proving that this is a metric!

1.4.1 Convergence

In a similar way, have you noticed that the definitions of convergence for a real/complex sequence are pretty much identical? Again, this is because we can generalise the definition of convergence to metric spaces!

Definition 1.5: (Convergence)
Let \((X,d)\) be a metric space, let \(a \in X\) and let \((a_n)_{n\in\mathbb{N}}\) be a sequence in \(X\). Then \((a_n)_{n\in\mathbb{N}}\) converges to \(a\) if \[\forall \epsilon > 0, \; \exists N = N(\epsilon) \in \mathbb{N}, \; \text{such that} \; \forall n \geq N, \; d(a_n,a) < \epsilon.\]

This is a huge generalisation of Definition 1.3, and you’ll meet it properly in Analysis 2A next year!

  1. For the time being, think of these functions as ones that you can draw without lifting your pen off the page.↩︎