Derivative of the Derivative?

Recently I had a very odd thought. Can we do calculus on functions that take and return functions (eg, (\mathbb{R} \to \mathbb{R}) \to (\mathbb{R} \to \mathbb{R}))?

Well, we’d need to have a conceptual idea of space between functions. Well, we could borrow one from the real numbers by constructing a trivial set of functions that are isomorphic to the real numbers: \{x \to kx | k \in \mathbb{R}\}.

Great. So we have an organized set of functions. It seems appropriate to call this a function space. Now we just need a functions that acts on it so we can do calculus.

Because it would be ironic, let’s choose \frac{d}{dx} as our function.

Hm… what happens when we apply \frac{d}{dx} to functions of the form x \to kx? \frac{d}{dx}(x \to kx)=x\to k.

Blast it! That’s outside our function space! What will we do? We’ll add another function space \{x \to k | k \in \mathbb{R}\}.

The cartesian product of those two earily similar to the second degree real polynomial space, P^2(\mathbb{R})… Let’s consider them to be the same thing!

So, it seems reasonable to say that the Jacobian of the n-dimensional version of this is:

J\left(\frac{d}{dx}\right) = \frac{d}{dx_{(1,2,3....)}}\left(\frac{d}{dx}\right)_{1,2,3...}= ~ \small \left[\begin{array}{cccc} 0&1&0&\ldots\\ 0&0&2&\ldots\\ 0&0&0&\ldots\\ \vdots&\vdots&\vdots&\ddots\\ \end{array}\right] ~~~ (in ~P^n(\mathbb{R})

Just some random thoughts. I mentioned this to a math professor and he said some stuff had been done on it in the field of functional analysis, but a quick search didn’t show much…

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