The name $\mathcal{C}^\omega$ Let A be a set (of real numbers); define $\mathcal{C}^\omega (A)$ as the set of all real-valued functions that are defined, bounded, and analytic on A.
My question is simply this: how did $\mathcal{C}^\omega (A)$ get its name?  Who named it?  What does $\omega$ mean in this context?  How does it extend the notion of $\mathcal{C}^n (A)$, where $n \in \mathbb{N} \cup \{0, \infty \}$?
Please cite.
 A: $\omega$ here is a symbol from logic corresponds to the ordinal number $\omega:=\cup_{n\to\infty} n$. $C^\omega$ is "more nice" as $C^\infty$, which I guess is what they wanted to get across and using another symbol for infinity seemed like a good way.
A: You ask two questions:

  
*
  
*Historically, how did this notation arise?
  
*Logically, what is the reasoning behind it?

I will answer Q2 only.
The reasoning is that we are meant to think of $\omega$ as being a number larger than $\infty$, as in:
$$0 \leq 1\leq 2\leq\cdots \infty\leq \omega$$
The nice thing about this is that it gives us the inclusions in the correct order:
$$\mathcal{C}^0(X) \supseteq \mathcal{C}^1(X) \supseteq \mathcal{C}^2(X)\cdots \mathcal{C}^\infty(X) \supseteq \mathcal{C}^\omega(X)$$
But honestly, I find this notation fairly lame. For starters, it should really be the other way around, with $\infty$ being the biggest element and $\omega$ the second-biggest. But that's just the start of the problems.
If you think about it for awhile, it becomes clear that what we should do is define an altogether new sequence $\mathcal{A}$ as follows: $\mathcal{A}^n(X)$ consists of all real-valued functions $f$ on $X$ such that for all $x \in X,$ there exists a neighbourhood of $x$ on which the function $f$ equals its own $n$th-order Taylor polynomial about $x$. So basically, $\mathcal{A}^n(X)$ consists of all functions on $X$ that piecewise (on each connected component of $X$) expressible as polynomials of degree $n$ or less (including the $0$ polynomial).
It therefore makes sense to write $\mathcal{A}^\infty(X)$ for the analytic functions on $X$.
With these conventions, we have the following system of inclusions:
$$\mathcal{C}^0(X) \supseteq \mathcal{C}^1(X) \supseteq \mathcal{C}^2(X)\cdots \mathcal{C}^\infty(X) \supseteq \mathcal{A}^\infty(X) \cdots \supseteq \mathcal{A}^2(X) \supseteq \mathcal{A}^1(X) \supseteq \mathcal{A}^0(X)$$
Of course, all the subscripts should really be at the bottom, as in $\mathcal{C}_n$ and $\mathcal{A}_n$.
