Let $K \subset \mathbb{R}$ be compact. For any function $f \in C^\infty(K, \mathbb{R}^n)$ how would one characterize the tangent space $T_f C^\infty(K, \mathbb{R}^n)$?
I am following a set of notes that says for any $f$, $T_f C^\infty(K, \mathbb{R}^n)$ can be identified with the space of smooth sections of some pullback bundle. If this is true, how would one arrive to this conclusion? Why do sections of the pullback bundle appear?
Secondly how would one define smooth paths in such a space?
If you want to make this all rigorous you need to define infinite-dimensional manifolds, which I'm not going to do, so this answer is not super rigorous; but hopefully it helps with your intuition.
I'd start with the paths. A smooth path in $X$ is just a smooth map $I \to X$ for $I \subset \mathbb R;$ so to define smooth paths in $X = C^\infty(K, \mathbb R^n)$ we just need to define what it means for a map $I \to X$ to be smooth. Since an element of $X$ is a curve $\gamma_0 : K \to \mathbb R^n,$ a path $I \to X$ is a family of curves $\gamma_t : K \to \mathbb R^n$ for $t \in I.$ This can interpreted as a function $\gamma : I \times K \to \mathbb R^n$ of two real variables, and the most straightforward notion of smoothness is that we require this function to be smooth, i.e. $\gamma \in C^\infty (I \times K, \mathbb R^n).$
Now that we have smooth paths, we can think about tangent vectors as the velocities of smooth paths. If $\gamma_t$ is a smooth path in $X$, its velocity is $$\dot\gamma_0(s) = \frac{\partial \gamma_t(s)}{\partial t}\Big|_{t=0}.$$ At each point $s \in K,$ this is a vector based at $\gamma_0(s) \in \mathbb R^k,$ so $\dot \gamma_0$ is a vector field along the curve $\gamma_0;$ i.e. a section of the pullback bundle $\gamma_0^*T\mathbb R^k.$ Thus we have $T_{\gamma_0}C^\infty(K, \mathbb R^k) = \Gamma(\gamma_0^*T\mathbb R^k).$