We know that in some cases, definitions and theorems might be swapped. That is, in some books, a property is used to define a concept, and then a second (equivalent) property is stated as a theorem, whilst in other books, the second property is stated as the definition and the first property is stated as a theorem.
Of course, a better word for phrase is "equivalent definitions".
But the situation can get more complicated than that. To give an example, the convolution product of two distributions could be defined in the following two ways:
- Firstly, for $u\in \mathcal D'(\mathbb R^n), \phi\in\mathcal D(\mathbb R^n)$ define $(u*\phi)(x)=\langle u(t),\phi (x-t) \rangle$. Then extend the definition to two distributions $u,v\in \mathcal D'(\mathbb R^n)$, one of which has a compact support.
- Use tensor products. $\langle u*v,\phi\rangle=\langle u\otimes v,\phi(x+y)\rangle$.
Since the theory of convolution products could take up more than 10 pages, using definition 1 to develop the theory will make the text very different from using definition 2 (and prove 1 as a theorem). If definition 1 is used, results in convolution product could be used to construct tensor product, so we can either construct the tensor product first and then construct the convolution product, OR the other way round: define convolution product first, then tensor product.
So there are quite a few possibilities if one is going to write a new book on this topic.
Are there any other examples of "definitions and theorems swapped" like this one? Perhaps, a better way to say this is: what are some examples of theories that could be developed from many different starting points?
My favorite example is normal subgroup. You can define it by left cosets are right cosets; or by $(aN)(bN)=(ab)N$ being well-defined; or by being the kernel of a group homomorphism. Then develop the whole theory of quotient groups and homomorphisms and the rest, starting from any one of the definitions, picking up the others as theorems along the way.