I've seen the notation $C^\omega$ used for the set of real analytic functions (e.g. on an interval). Where does it come from? What exactly does it mean? What is the reason behind it? Who first used it?
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The idea of the notation is that $\omega$ is the ordinal following all the finite ordinals. It is larger than all of them: being analytic is a bit more than being differentiable to all orders.
I doubt one can trace who first used the notation, really. (Is it at all interesting?)