Do these properties characterize differentiation?

Question

Let $L: C^\infty(\mathbb{R}) \to C^\infty(\mathbb{R})$ be a linear operator which satisfies:

$L(1) = 0$

$L(x) = 1$

$L(f \cdot g) = f \cdot L(g) + g \cdot L(f)$

Is $L$ necessarily the derivative? Maybe if I throw in some kind of continuity assumption on $L$? If it helps you can throw the "chain rule" into the list of properties.

I can see that $L$ must send any polynomial function to it's derivative. I want to say "just approximate any function by polynomials, and pass to a limit", but I see two complications: First $\mathbb{R}$ is not compact, so such an approximation scheme is not likely to fly. Maybe convolution with smooth cutoff functions could help me here. Even if I could rig up something I am concerned that if polynomials $p_n$ converge to $f$, I may not have $p_n'$ converging to $f'$. My Analysis skills are really not too hot so I would like some help.

I am interested in this question because it is a slight variant of a characterization given here:

Why do we teach calculus students the derivative as a limit?

I am not sure whether or not those properties characterize the derivative, and they are closely related to mine.

If these properties do not characterize the derivative operator, I would like to see another operator which satisfies these properties. Can you really write one down or do you need the axiom of choice? I feel that any counterexample would have to be very weird.

Answer 1

Yeah, these force it to be ordinary differentiation. We have to show that for each fixed $x_0 \in \mathbb{R}$, the composite

$$C^\infty(\mathbb{R}) \stackrel{L}{\to} C^\infty(\mathbb{R}) \stackrel{ev_{x_0}}{\to} \mathbb{R}$$

is just the derivative at $x_0$. For each $f \in C^\infty(\mathbb{R})$, there is a $C^\infty$ function $g$ such that

$$f(x) = f(x_0) + f'(x_0)(x - x_0) + (x - x_0)^2g(x)$$

and so $(ev_{x_0} L)(f) = ev_{x_0}(f'(x_0) + 2(x - x_0)g(x) + (x - x_0)^2 L(g)(x))$ by the properties you listed. Of course evaluation at $x_0$ kills the last two terms and one is left with $f'(x_0)$, as desired.

Answer 2

Here is a slightly more general take on this question. First notice that your condition $L(1) = 0$ is redundant. That is because if you take $f = g = 1$ in your third condition, you get $L(1) = L(1) + L(1)$. It is a theorem that goes back as far as I know to Chevalley (see around page 76 in his Theory of Lie Groups) that if $M$ is a $C^\infty$ manifold, then any linear map $L$ of $C^\infty(M)$ to itself that satisfies the derivation condition (your third condition) is a smooth vector field. This means that in local coordinate $(x_1,\ldots,x_n)$ it has the form $L(f) = \sum_i h_i {\partial f\over \partial x_i}$ where the $h_i$ are smooth functions. Moreover, if we compute $L(x_j)$ using this formula we see that $h_j$ is just $L(x_j)$. So in particular if a linear map $L$ of $C^\infty (R^n)$ to itself satisfies the derivation condition and $L(x_i) = \delta_{ij}$, then $L = {\partial \over \partial x_j}$.

Answer 3

I remember thinking the same thing as Richard is saying and thinking about defining $L$ simply by using the third property. I believe only using this property you can prove change rule (but maybe I'm wrong, and my notes are far away).

The caveat is that if you don't use $L(x)=1$, then you can think of (with suitable restrictions on domains) the following:

$$L(f)(x)=\frac{f(x)}{x}$$

which, by the way, satisfies the chain rule. :)