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I am currently studying for a first course in operator theory, and I was wondering about the following that occurred to me whilst doing my reading:

It seems to me that, by talking about maps in general we have a process that allows us to change from very "basic" mathematical objects, such as a point, to more and more complex objects.

For example:

Suppose $n$ is a natural number. We can think of the set $\mathbb{R}$ as the vector space of all the possible (continuous) functions $n \mapsto x_n \in \mathbb{R}$. In the next step we have the vector space $C(\mathbb{R})$ of all possible continuous functions $\mathbb{R} \ni x \mapsto f(x)$. And then there is the vector space $\mathcal{C}(C(\mathbb{R}))$ of all possible continuous operators $C(\mathbb{R}) \ni f \mapsto Af$.

So in each step we take the previous object and think about all possible continuous functions on it. The thing I was wondering about is - what are the objects that we use in the next step ? That is, what objects act on the space $\mathcal{C}(C(\mathbb{R}))$ ?

\begin{equation} n \to \mathbb{R} \to C(\mathbb{R}) \to \mathcal{C}(C(\mathbb{R})) \to \, ? \end{equation}

I realize this question might be very stupid, I hope it makes sense at least .. appologies in case not, I am just starting to learn about operators. Thanks for your help !

EDIT: the notation $\mathcal{C}(C(\mathbb{R}))$ might not be the one that is used commonly, so far I have only seen linear operators and their common notation, in case somebody could hint towards the proper notation that would be great ! I shall also have a look at books to find out, but until that I appologize if the notation looks extravagant

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    @martini I have edited the question accordingly so that it is now more clear (hopefully), my initial restriction to linear operators was due to the fact that I am just a beginner in operator theory, so I haven't seen anything on non - linear operators hence I was a little scared to mention them. But it seems it makes the question unclear, as Florian already mentioned this issue2012-04-12

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When you say "continuous functions on $C(\mathbb{R})$", you don't say what topology you are taking. The most natural choice is probably uniform convergence (because it makes it complete); but then you get a topological space which is not particularly nice, as it is not locally compact. And I don't think much can be said about the set of continuous functions over a non-LC space.