There are three ways to define when a ($\mathbb{R}$-valued) function on a closed interval is smooth:

- $f$ can be extended to a smooth function on $(a - \epsilon, b + \epsilon)$ for some $\epsilon > 0$
- $f \in \bigcap_{k \geq 0} C^k([a, b])$. Here $f \in C^1([a, b])$ when the limit of the differences quotients of $f$ is exists at every point and defines a continuous function on $[a, b]$ and $f \in C^k([a, b])$ if $f' \in C^{k-1}([a, b])$. $\label{Ck}$
- $f$ is smooth on $(a, b)$ and all of its derivatives extend continuously to $[a, b]$

Obviously $(1) \Rightarrow (2) \Rightarrow (3)$ and to show $(3) \Rightarrow (1)$ one can use that it's possible to construct smooth functions with arbitrary Taylor series at some point, as is for example done in [1, Lemma A.1.3].

I'm interested in the case where the function takes values in some kind of topological vector space $E$. A derivative of $f$ can still be defined as the limit of difference quotients.

**Q: For what kind of TVSs do some of these three definitions coincide?**

The cited constructions uses an absolutely and uniformly convergent series to define a smooth extension of function satisfying $(3)$, so I think it still works whenever $E$ is a Banach space.

I'm most interested in the case where $E$ is a Fréchet space.

**Reference:**