## 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…