Example of a linear map over $\mathbb R(X)$ which isn't a linear map over $C^1$

Question

Over $\mathbb R$, the only linear maps are those of the form $ax$.

If we discuss rational functions over $\mathbb R$, this extra structure would allow us to describe a wider variety of linear maps.

But the obvious maps such as limits, differentiation, summation, doing $f(x)\mapsto f(x+k)$ seem too easy. Is there a linear map which specifically takes advantage of the fact that we have a rational function?

For instance, something like "double the coefficient of $x$ of the numerator and triple that of $x$ in the denominator". I know this is not a good example because it isn't linear, but it illustrates what I mean by "taking advantage of the fact that we have a rational function", i.e., something which doesn't generalise easily to a wider class of functions (say $C^1$).

Answer 1

Any linear map $T$ of vector spaces $X \to Y$ can be "generalized" to a map of $Z \to Y$, where $Z$ is any vector space that contains $X$. Namely, let $P$ be a projection from $Z$ onto $X$, and take the linear map $T \circ P$. Of course, you might not be able to obtain $P$ explicitly: you might need the Axiom of Choice.

Answer 2

Using partial fractions and the fact that $\mathbb{R}[X]$ is a PID, it's possible to write every element of $\mathbb{R}(X)$ uniquely as a sum $\sum_{p<\mathbb{R}[X],n\in\mathbb{N}} \frac{f_{p,n}}{p^n}$, where $p$ ranges over the primes of $\mathbb{R}[X]$ and $f_{p,n}$ has smaller degree than $p$.

This expresses $\mathbb{R}(X)$ as an infinite direct sum of finite dimensional vector spaces (namely vector spaces of dimension $1$ for the linear primes and dimension $2$ for indecomposable quadratics), so we can apply any linear maps we like independently to each piece of the decomposition. Most of these maps will be strange and have no obvious generalization to arbitrary continuous functions.

Answer 3

A example of a linear functional on $\mathbb R(X)$ ... write it in partial fraction form, and take the coefficient of $1/(X-3)^2$. (Of course $0$ if that terms does not appear.)