Taylor Series as a linear operator $T:C^{k} (\mathbb{R} , \mathbb{R}) \to C^{\infty} (\mathbb{R} , \mathbb{R})$? Can the Taylor series be thought of as either a linear operator $T: C^{k} (\mathbb{R} , \mathbb{R}) \to C^{\infty} (\mathbb{R} , \mathbb{R})$  given by
$$
Tf=\sum^{k}_{n=0} \frac{f^{(n)}(a)}{n!}(x-a)^{n} 
$$
for $f \in C^{k} (\mathbb{R} , \mathbb{R})$ with the special case  $T:C^{\infty} (\mathbb{R} , \mathbb{R}) \to C^{\infty} (\mathbb{R} , \mathbb{R})$ given by 
$$
Tf =\sum^{\infty}_{n=0} \frac{f^{(n)}(a)}{n!}(x-a)^{n}$$
for a function $f \in C^{\infty}(\mathbb{R} , \mathbb{R})$? The idea just popped into my head and I wanted to make sure I wasn't going down the rabbit hole.
 A: This works for finite $k$ (and a fixed point $a$).  But for $k=\infty$, it doesn't work because the sum $\sum^{\infty}_{n=0} \frac{f^{(n)}(a)}{n!}(x-a)^{n}$ may not converge.  For instance, if $(c_n)$ is any sequence of real numbers, then there exists a $C^\infty$ function $f:\mathbb{R}\to\mathbb{R}$ such that $f^{(n)}(0)=c_n$ for each $n$.  If you choose the $c_n$ to grow fast enough (e.g., $c_n=(n!)^2$), then $\sum^{\infty}_{n=0} \frac{c_n}{n!}x^{n}$ will not converge for any $x\neq0$.
A: I'm going to use this answer to address the concern that AnalysisStudent expressed about "going down the rabbit hole," possibly made precise by Roland in the comments:

Is this an interesting operator?

Without digging too much into what it means to be an interesting operator, I'll just say that I think the answer is "yes." To motivate that, I'm going to describe a situation where operators similar to this are interesting.
In the numerical approximation of solutions to differential equations, one wishes to obtain methods of producing approximate solutions to differential equations. Such a method often involves parameters, such as geometric approximations, as well as computational limits such as approximating real numbers by floats. 
Given such a method, one is concerned with the fundamental questions: As I vary the parameters in the method (for example, make an approximating grid better and better) does the method converge to an actual solution? And, if I have a particular solution arrived at with fixed parameters, how close is it to a the actual solution? These questions are answered with proofs, and underpin confidence in the numerical solutions we arrive at.
Looking at these proofs, I've found it useful to conceptually separate two types of approximations. The first is the approximation of real numbers by floating-point numbers. The second is the approximation of smooth ($C^\infty$) functions by "easily computable" functions. By that, I mean, functions which translate an infinite-dimensional problem, like finding the eigenvalues of a differential operator, into a finite dimensional problem, like finding the eigenvalues of a large matrix.
So the process goes like this:
$$ \mbox{equation in $C^\infty$} \xrightarrow{2^{nd} approx.} \mbox{equation in $\Bbb{R}^n$} \xrightarrow{1^{st} approx.} \mbox{numerical solution} $$
that is
$$ \mbox{functional equation} \to \mbox{matrix equation} \to \mbox{numerical analysis} $$
Considerations like floating point errors, numerical stability, and well- and ill-conditioned go with the first type of approximation. Considerations like the operator in the original question go with the second. Let's focus on those.
In reducing a differential equation to a linear algebra problem, we may want to replace an infinite dimensional function space with a finite dimensional subspace. The goal is to choose a finite dimensional subspace where the problem becomes manageable with finite computing power, and then numerically solve it. 
If we have a function space $\mathcal{V}$ and a finite dimensional subspace $V\subset\mathcal{V}$ which we believe is suitable for studying and approximating the problem, then the next step is to ask how we translate the functions from $\mathcal{V}$ to $\mathcal{V}$. The answer is: via an operator like $T$ from the original question. Once we have this operator, we want to study all sorts of things about it --- how it distorts distances (i.e. how good it is at approximating), what kind of functions are in its kernel or close to its kernel, what kinds of functions are badly approximated, how it interacts with the problem we want to solve, and so on.
To make this concrete, say we have a triangular mesh on a polygonal Euclidean domain and we want to study how well piecewise-polynomial interpolation on the mesh approximates smooth functions on the domain. We do this by studying triangle-by-triangle and examining how well piecewise-polynomial interpolation on a single triangle works for smooth functions. (One answer is given by the Bramble-Hilbert lemma; other answers in different norms are given by studying the remainder operator, $R = T-I$ in the notation of the question.) Once this is known, the error can be summed over triangles to produce estimates that make their way into the error estimates on various approximations of the solutions of linear partial differential equations on the domain.
I'm being very vague here, but I hope this provides some context and some justification for why questions like this are interesting to some people.

As an aside, the philosophy "choose a good space!" is also true in the the study of the solutions of differential equations. This is where spaces like the $L^p$ spaces, the Sobolev spaces $W^{k,p}$ and $H^k$, the $BMO$ spaces, and so on come from: People found them useful contexts for studying various types of functional equations, identifying solutions, and studying the properties of the solutions.
