Taylor Series as a linear operator $T:C^{k} (\mathbb{R} , \mathbb{R}) \to C^{\infty} (\mathbb{R} , \mathbb{R})$?

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 $x\neq0$.