An example for example

Tim Gowers has two very interesting posts on using examples early on in a mathematical exposition of a subject. I can only second that and say that this is my favorite way of understanding mathematical concepts: Try to think through the simplest non-trivial example.
Of course, for a mathematician it could be enough just to state a definition or state a theorem (including a proof) but very often this leaves one without a proper understanding of the subject. Why this definition and not something else? Where and how can I use the theorem? Why do I have to make assumption x, y and z?

For my practical purposes I often want to see "the key example" that shows what's going on and then often the general theory is a more or less obvious generalisation or extension or formalisation or abstraction of this key example. This second step hopefully is then clear enough that one can come up with it by oneself and it's really the key example one should remember and not the formal wording of the definition/theorem etc.

I am not talking about those examples some mathematicians come up with when pressed for an example, like after stating the definition of a vector space giving {0} as the example. This is useless. I want to see a typical example, one that many (all) other cases are modeled on not the special one that is different from all other cases. And as important as examples are of course counter-examples: What is close but not quite according to the new definition (and why do we want to exclude it? What goes wrong if I drop some of the assumptions of the theorem?

I have already talked for too long in the abstract, let me give you some examples:

• What's a sheaf and what do I need it for (at least in connection with D-branes)? Of course, there is the formal definition in terms of maps from open sets of a topological space into a category. The wikipedia article Sheaf reminds you what that is (and explains many interesting things. I think I only really understood what it really is after I realised that it's the proper generalisation of a vector bundle for the case at hand: A vector bundle glues some vector space to every point of a topological space and does that in a continuous manner (see, that's basically my definition of a vector bundle). Of course, once we have such objects, we would like to study maps between them (secretly we want to come up with the appropriate category). We know already what maps between vector-spaces look like. So we can glue them together point-wise (and be careful that we are still continuous) and this gives us maps between vector bundles. But from the vector-space case we know that then a natural operation is to look at the kernel of such a map (and maybe a co-kernel if we have a pairing). We can carry this over in a point-wise manner but, whoops, the 'kernel-bundle' is not a vector bundle in general: The dimension can jump! The typical example here is to consider as a one dimensional vector bundle over the real line (with coordinate x). Then multiplication in the fiber over the point x by the number x is trivially a fiber-wise linear map. Over any point except x=0 it has an empty kernel but over x=0 the kernel is everything. Thus, generically the fiber has dimension 0 but at the origin it has dimension one. Thus, in order to be able to consider kernels (and co-kernels) of linear bundle maps we have to weaken our definition of vector bundle and that what is a sheaf: It's like a vector bundle but in such a way that linear maps all have kernels and co-kernels.
• When I was a student in Hamburg I had the great pleasure to attend lectures by the late Peter Slodowy (I learned complex analysis from him as well as representation theory of the Virasoro algebra, gauge theories in the principle bundle language, symplectic geometry and algebraic geometry). The second semester of the algebraic geometry course was about invariants. Without the initial example (which IIRC took over a week to explain) I would have been completely lost in the algebra: The crucial example was: We want to understand the space of matrices modulo similarity transformations . Once one has learned that the usual algebraic trick is to investigate a space in terms of the algebra of functions living on it (as done in algebraic geometry for polynomial functions, or in non-commutative geometry in terms of continuous functions) one is lead to the idea that this moduli space is encoded in the invariants, that is functions that do not change under similarity transformations. Examples of such functions are of course the trace or the determinant. It turns out that this algebra of invariants (of course the sum or product of two invariant functions is still invariant) is generated by the coefficients of the characteristic polynomial that is, by the elementary symmetric functions of the eigenvalues (eigenvalues up to permutations). So this should be the algebra of invariants and its dual the moduli space. But wait, we know what the moduli space looks like from linear algebra: We can bring any matrix to Jordan normal form and that's it, matrices with different Jordan normal forms are not similar. But both and have the same characteristic polynomial but are not related by a similarity transformation. In fact the second one is similar to any matrix
  for any . This shows that there cannot be a continuous (let alone polynomial) invariant which separates the two orbits as the first orbit is a limit point of points on the second orbit. This example is supposed to illustrate the difference between which is the naive space of orbits which can be very badly behaved and the much nicer which is the nice space of invariants.
• Let me also give you an example for a case where it's hard to give an example: You will have learned at some point that a distribution is a continuous linear functional on test-functions. Super. Linear is obvious as a condition. But why continuous? Can you come up with a linear functional on test-functions which fails to be continuous? If you have some functional analysis background you might think "ah, continuous is related to bounded, let's find something which is unbounded". Let me assure you, this is the wrong track. It turns out you need the axiom of choice to construct an example (as in you need the axiom of choice to construct a set which is not Lebesgue measurable). Thus you will not be able to write down a concrete example.
• Here is a counter-example: Of course is the typical example of a real finite dimensional vectors space. But it is very misleading to think automatically of when a real vector space is mentioned. People struggled long enough to think of linear maps as abstract objects rather than matrices to get rid of ad hoc basis dependence!

I a sitting in the "Mathematical quantum mechanics" class of our TMP program. There, to my mind, Prof. Siedentop does a pretty good job of motivating the ideas (in terms of examples and 'engineer talk' --- he means physicists talk) behind the definitions and proofs that he presents rather than getting bogged down in the technical details. Let's hope our (the physicists) insistence on this background information does not make him lose so much time he does not get to where he wants in the end...