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

  1. $f$ can be extended to a smooth function on $(a - \epsilon, b + \epsilon)$ for some $\epsilon > 0$
  2. $f \in \bigcap_{k \geq 0} C^k([a, b])$. 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 $f' \in C^{k-1}([a, b])$. $\label{Ck}$
  3. $f$ 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:

  1. McDuff and Salamon, Introduction to Symplectic Topology, 3e, Oxford Graduate Texts in Mathematics
    Here is the relevant Lemma Introduction to Symplectic Topology, Lemma A.1.3 on page 575
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  • $\begingroup$ Doesn't that make 3 ways ? $\endgroup$ – Loïc Teyssier Nov 12 '20 at 12:47
  • $\begingroup$ @LoïcTeyssier Thanks, I fixed it (I originally only had (1) and (3) in mind) $\endgroup$ – Carlos Esparza Nov 12 '20 at 18:49
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The lemma is a proof for the so-called Borel theorem from 1895 although it was independently shown about 10 years earlier by Peano. It is a very special case of Whitney's extension theorem.

I think that the answer to your question follows from the usual tensor representation of Frechet valued smooth functions: $C^\infty(\mathbb R,E)=C^\infty(\mathbb R) \hat\otimes_\pi E$ and $C^\infty([a,b],E)=C^\infty([a,b]) \hat\otimes_\pi E$ (I did not check for references, this is probably already in Grothendieck's thesis). In the scalar case, the restriction map $r:C^\infty(\mathbb R)\to C^\infty([a,b])$ is surjective (even more, by a theorem of Mityagin and independently Seeley, it has a continuous linear right inverse) and the $\pi$-tensor product of Frechet spaces respects quotients and therefore the Frechet valued restriction mapping $R=r\otimes id_E$ is also surjective.

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(3) $\Rightarrow$ (2) and hence (2) $\Leftrightarrow$ (3) whenever $E$ is any Hausdorff locally convex space. This is a simple consequence of the vector valued mean value theorem that in turn by Hahn−Banach follows easily from the result for real valued functions on a closed real interval.

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