I’m mildly obsessed with examples. Whenever I feel confused, it’s often because I don’t know enough examples. Proofs can be confusing too, if some step is poorly explained. But that kind of confusion tends to be local. You can still have a good grasp of the theory. If you don’t know enough examples, you feel generally lost. Textbooks lacking in examples end up being dry. In contrast, books with well-chosen examples are a pleasure to read¹.

According to the Cambridge Dictionary, an example is “something that is typical of the group of things that it is a member of”.

In mathematics, there are many kinds of examples, and they all serve different purposes. Traditionally, fields like topology and measure theory place a greater emphasis on counter examples. In algebra, there are more prototypical examples; definitions are often followed by an exhaustive list of objects of that category². In the context of mathematics, the dictionary definition is too simplistic! So here is an attempt at classifying the main kinds of examples, or, alternatively, an examples appreciation post.

Understanding definitions #

First, there are the examples helping us understand definitions.

• Prototypical examples: examples just complex enough to capture the essential properties of an object.
  • Fundamental groups: the fundamental group of the circle $\pi_1(\mathbb{S}^1, x_0)$.
  • Transcendental field extensions: $\mathbb{Q}(\pi)$.
  • Modular forms: the Poincaré series $P_{m, k}(z)$.
• Canonical examples: the canonical construction of something - the construction you get which involves the least arbitrariness.
  • Stochastic processes: the coordinate process.
  • Embeddings into bi-duals: take the embedding $V \to V^{**}$ sending $v$ to the evaluation.
  • Rings: the integers $\mathbb{Z}$.
• Foundational examples: examples of special interest and which form the basis for further theory.
  • Anything analysis: the Gaussian $\phi(x) = e^{-x^2/2}$.
  • Stochastic processes: the simple random walk on $\mathbb{Z}$.
  • Modular forms: the $j$–invariant.
• Basic examples: the simplest possible instance of an object. The kind of example an uncreative student might cite in an exam³.
  • Groups: the trivial group $G := \{e\}$.
  • Martingales: the martingale $(X_t)_{t \ge 0}$ where $X_t \equiv 1$ for all $t$.
  • Banach space: just take $\mathbb{R}$.
• Non-examples: objects which are not instances of something.
  • A non-ring: the natural numbers $\mathbb{N}$.
  • A non-tempered distribution: the function $e^t$.
  • A non-Artinian ring: the integers $\mathbb{Z}$.
• Computational examples: examples involving a computation and that illustrate how to work with a given object.
  • Compute $\mathbb{R}[x] \otimes \mathbb{C}$.
  • Find the Euler product expansion for the Dirichlet generating series of the Möbius function $\mu = 1^{-*}$
  • Check that $c^{-1/2} W_{ct}$ is again Brownian motion.
• Pathological examples: examples illustrating distinctions between different notions, or the limitations of a certain concept.
  • Differentiability $\neq$ continuity: consider the Weierstrass function.
  • Connectedness $\neq$ path connectedness: because the Topologist’s sine curve.
  • Lebesgue integral $\neq$ Riemann integral: try integrating the Dirichlet function, the characteristic function of the rationals.
• Toy examples: examples you can manipulate to have some desired properties. When looking for counter examples, you typically start with some toy example.
  • Random variable: define $$X := \begin{cases}\alpha, & \text{with probability } p, \\\ \beta, & \text{with probability } 1 - p.\end{cases}$$
  • Measure theory: simple functions.
  • Galois groups: consider $\mathrm{Gal}(\mathbb{Q}(\sqrt{\alpha}):\mathbb{Q})$.
• Real-world examples: a real-world scenario involving the object of interest. These help motivate the study of the given object and are useful for gaining intuition.
  • Stochastic processes: a Poisson process as a description of the number of raindrops falling in a given square.
  • PDE: the heat equation.
  • Graph theory: a network models the flow of some fluid.

Understanding results #

Second, there are the examples helping us understand results. I’ll count applications of theorems among these kinds of examples.

• Counter examples: examples illustrating how a given result breaks down if we drop assumptions⁴.
  • Fatou’s lemma: we really do need non-negativity. Consider the sequence $f_n := - 1_{[n, n+1]}$.
  • Open mapping theorem: surjectivity is necessary, since the zero map isn’t open.
  • Nullstellensatz: the field $K$ must be algebraically closed. For $K = \mathbb{R}$, the ideal $I = (x^2 + 1)$ is maximal but not of the form $(x - a)$.
• Basic applications: checking that the conditions of a theorem are satisfied, apply the theorem and see what you get. These examples are particularly useful if the theorem statement seems involved.
  • Hurewicz’ theorem: use it to find $H_1(\mathbb{S}^n)$.
  • Dedekind-Kummer: compute the factorisation of $(p)$ in a ring of integers.
  • Dirichlet’s unit theorem: the theorem allows us to verify that $\mathbb{Q}(\sqrt{d})$ has finite unit group.
• Non-basic applications: or corollaries. Deduce something interesting from a theorem.
  • Optional stopping: allows you to compute laws of hitting times.
  • The ring of integers is Dedekind: this gives us a satisfying proof of Fermat’s theorem for primes that are sums of squares.
  • Hausdorff-Young: an application of Riesz-Thorin tells us that the Fourier transform defines a bounded linear operator from $L^p$.
• Amusing applications: examples mentioned during lectures for amusement.
  • Borsuk-Ulam: there are two antipodal points on earth with the same temperature.
  • Mean-value theorem: you can catch someone over-speeding using the mean-value theorem.
  • Four colour theorem: you can colour a map using just four colours so no two adjacent countries have the same colour.
• Satisfying applications: prove a well-known theorem as an application of the fancy theory you developed.
  • The insolvability of the quintic: “just” an example application of Galois theory.
  • The central limit theorem: falls out of the computation of the characteristic function of the scaled sum.
  • The fundamental theorem of algebra: an application of Liouville’s theorem.

Are examples the core of mathematics? The answer is very much a matter of personal taste. I’d say yes. Apparently von Neumann once said

“In mathematics you don’t understand things. You just get used to them.”

There’s some truth to this. And the way we get used to things is by studying examples.

Thanks to Alois Schaffler for suggesting the last kind of example.