The name $\mathcal{C}^\omega$

Question

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.

Answer 1

$\omega$ 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.

Answer 2

You ask two questions:

1. Historically, how did this notation arise?

2. 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 $\mathcal{A}_n$.