Why not number some definitions in Rudin's book?

Question

I'm reading Rudin's Principle of Math Analysis

Why some definitions about functions in Chapter 8, there are no the number marks, and the boldfaced words Definition?

This chapter only have 22 marked theorem and definitions and...

For example

Section 3

We define

$$\begin{align*}E(z)=\sum _{n=0}^{\infty } \frac{z^n}{n!}\quad \quad\tag{1}\end{align*}$$

Section 4

Let us define

$$\begin{align*}C(x)=\frac{1}{2}[E(\text{ix})+E(-\text{ix})]\tag{2}\end{align*}$$

Is there some reason's in doing these things? i.e. Boldface and number some theorems/definitions in one book.

I suggest you ask Rudin (is he still with us?). Sorry, he passed away in 2010. I guess we'll never know. — Gerry Myerson, Jul 21 '13 at 12:51
@GerryMyerson fine...Maybe this is one general question? Other's books have some similar situations? — HyperGroups, Jul 21 '13 at 12:53
Maybe it's more a notation than really the definition of a concept. — Davide Giraudo, Jul 21 '13 at 12:54
I'm not sure what the exact question is, but there's a number besides each definition/formula. That seems to be enough. — DonAntonio, Jul 21 '13 at 13:04

score 3 · Accepted Answer · answered Jul 21 '13 at 13:00

These are purely local definitions, used only temporarily. The whole point of those sections is to prove that this function $E(z)$ is just the exponential function $e^z$, that this $C(x)$ is just $\cos x$, and that the similarly defined $S(x)$ is just $\sin x$.

Why not number some definitions in Rudin's book?

Section 3

Section 4

1 Answers1