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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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This does not look convincing to me at all. The passage from $C(\mathbb{R})$ to $\mathcal{B}(C(\mathbb{R}))$ is not that much analogous than the one from $\mathbb{R}$ to $C(\mathbb{R})$, because one case is concerned with linearity and the other one isn't. And I don't understand what you mean by $n\to \mathbb{R}$ or $n\mapsto \mathbb{R}$. –  Florian Apr 12 '12 at 11:49
    
@Florian: For any singleton, e.g. $\{n\}$, there is an obvious correspondence between $\Bbb R$ and the set of maps from the singleton to $\Bbb R$; that’s what harlekin is talking about for his first step. –  Brian M. Scott Apr 12 '12 at 12:03
    
@Florian: thanks for your comment! The map you wondered about looked strange indeed, what Brian M. Scott suggests is what I wanted to write. I edited the post accordingly. With regards to the second step - you are right the process looses generality in that it restricts to linear maps (operators) on the preceeding space. I could probably reduce the post to the question "what are the maps that act on bounded linear operators ?" .. –  harlekin Apr 12 '12 at 12:20
    
Well, once you have a vector space, you can always take the dual. But this may not be what you are looking for. –  M Turgeon Apr 12 '12 at 12:43
    
@M Turgeon: that's a very good point - though the dual would then "collapse" everything back to $\mathbb{R}$ (roughly speaking). I was rather looking for a space of maps $\mathcal{B}(C(\mathbb{R})) \to \mathcal{B}(C(\mathbb{R})) $. But in any case, what you say partly answers the question, thanks for pointing this out! –  harlekin Apr 12 '12 at 13:15
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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.

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